Built with Alectryon, running Coq+SerAPI v8.10.0+0.7.0. Coq sources are in this panel; goals and messages will appear in the other. Bubbles () indicate interactive fragments: hover for details, tap to reveal contents. Use Ctrl+↑ Ctrl+↓ to navigate, Ctrl+🖱️ to focus.
(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

Require Import Rbase.
Require Import Rfunctions.
Require Import Rseries.
Require Import SeqProp.
Require Import PartSum.
Require Import Max.

Local Open Scope R_scope.

(***************************************************)
(* Various versions of the criterion of D'Alembert *)
(***************************************************)


forall An : nat -> R, (forall n : nat, 0 < An n) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0 -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

forall An : nat -> R, (forall n : nat, 0 < An n) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0 -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0

({l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}) -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

bound (EUn (fun N : nat => sum_f_R0 An N))
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2

exists m : R, is_upper_bound (EUn (fun N : nat => sum_f_R0 An N)) m
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2

exists m : R, is_upper_bound (EUn (fun N : nat => sum_f_R0 An N)) m
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2

is_upper_bound (EUn (fun N : nat => sum_f_R0 An N)) (sum_f_R0 An x + 2 * An (S x))
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat

sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat

sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat

sum_f_R0 An x1 <= sum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat
sum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) = sum_f_R0 An x
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat

0 <= sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat
sum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) = sum_f_R0 An x
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat

0 < sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat
0 < 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat
sum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) = sum_f_R0 An x
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat

0 < 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat
sum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) = sum_f_R0 An x
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x1 < x)%nat

sum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) = sum_f_R0 An x
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2

sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2

0 <= 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat

sum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat

sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= sum_f_R0 An x + 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat

sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat

sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x) -> sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x) <= 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x) <= 2 * An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

0 <= An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x) <= 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x) <= 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

(1 - (/ 2) ^ S (x1 - S x)) / (1 - / 2) <= 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

(1 - (/ 2) ^ S (x1 - S x)) / / 2 <= 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 = 1 - / 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

(1 - (/ 2) ^ S (x1 - S x)) * 2 <= 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 = 1 - / 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

0 <= 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
1 - (/ 2) ^ S (x1 - S x) <= 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 = 1 - / 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

1 - (/ 2) ^ S (x1 - S x) <= 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 = 1 - / 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

(/ 2) ^ S (x1 - S x) + (1 - (/ 2) ^ S (x1 - S x)) < (/ 2) ^ S (x1 - S x) + 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 = 1 - / 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

1 < (/ 2) ^ S (x1 - S x) + 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 = 1 - / 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

0 < (/ 2) ^ S (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 = 1 - / 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 = 1 - / 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

/ 2 = 1 - / 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

2 * / 2 = 2 * (1 - / 2)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

1 = 2 * 1 - 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

2 <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
H3:sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)

/ 2 <> 1
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat

sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x) * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat

sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= An (S x + 0)%nat * sum_f_R0 (fun i : nat => (/ 2) ^ i) (x1 - S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat

0 <= / 2
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
forall i : nat, An (S x + S i)%nat < / 2 * An (S x + i)%nat
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat

forall i : nat, An (S x + S i)%nat < / 2 * An (S x + i)%nat
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
i:nat

(forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An n) -> An (S x + S i)%nat < / 2 * An (S x + i)%nat
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
i:nat
forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An n
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
i:nat
H4:forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An n

An (S (S x + i)) < / 2 * An (S x + i)%nat
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
i:nat
H4:forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An n
S (S x + i) = (S x + S i)%nat
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
i:nat
forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An n
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
i:nat
H4:forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An n

S (S x + i) = (S x + S i)%nat
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
i:nat
forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An n
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
i:nat

forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An n
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat

0 < / An n
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat
/ An n * An (S n) < / An n * (/ 2 * An n)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat

/ An n * An (S n) < / An n * (/ 2 * An n)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat

An (S n) * / An n < / 2 * 1
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat
An n <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat

Rabs (Rabs (An (S n) / An n) - 0) < / 2
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat
Rabs (Rabs (An (S n) / An n) - 0) = An (S n) * / An n
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat
An n <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat

Rabs (Rabs (An (S n) / An n) - 0) = An (S n) * / An n
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat
An n <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat

An (S n) / An n = An (S n) * / An n
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat
An (S n) / An n >= 0
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat
An n <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat

An (S n) / An n >= 0
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat
An n <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n0 : nat, 0 < An n0
H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2
x1:nat
l:(x < x1)%nat
i, n:nat
H3:(n >= x)%nat

An n <> 0
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat

An (S x + 0)%nat = An (S x)
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat
sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:0 < / 2
x:nat
H2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2
x1:nat
l:(x < x1)%nat

sum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0

{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
x:R
H1:is_lub (EUn (fun N : nat => sum_f_R0 An N)) x

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x; apply Un_cv_crit_lub; [ unfold Un_growing; intro; rewrite tech5; pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; apply H | apply H1 ]. Defined.

forall An : nat -> R, (forall n : nat, An n <> 0) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0 -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

forall An : nat -> R, (forall n : nat, An n <> 0) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0 -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R

(forall n : nat, 0 < Vn n) -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n

(forall n : nat, 0 < Wn n) -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n

Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0 -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0

Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0 -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:Un_cv (fun N : nat => sum_f_R0 Vn N) x

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:Un_cv (fun N : nat => sum_f_R0 Vn N) x
x0:R
p0:Un_cv (fun N : nat => sum_f_R0 Wn N) x0

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0

0 < eps / 2 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 An n) (x - x0) < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n : nat, (n >= x1)%nat -> R_dist (sum_f_R0 Vn n) x < eps / 2

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 An n) (x - x0) < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x, x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n : nat, (n >= x1)%nat -> R_dist (sum_f_R0 Vn n) x < eps / 2

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 An n) (x - x0) < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x, x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n : nat, (n >= x1)%nat -> R_dist (sum_f_R0 Vn n) x < eps / 2
x2:nat
H9:forall n : nat, (n >= x2)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps / 2

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 An n) (x - x0) < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n : nat, (n >= x1)%nat -> R_dist (sum_f_R0 Vn n) x < eps / 2
x2:nat
H9:forall n : nat, (n >= x2)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps / 2

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 An n) (x - x0) < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n : nat, (n >= x1)%nat -> R_dist (sum_f_R0 Vn n) x < eps / 2
x2:nat
H9:forall n : nat, (n >= x2)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps / 2
N:=Nat.max x1 x2:nat

exists N0 : nat, forall n : nat, (n >= N0)%nat -> R_dist (sum_f_R0 An n) (x - x0) < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat

R_dist (sum_f_R0 Vn n - sum_f_R0 Wn n) (x - x0) < eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
sum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat

Rabs (sum_f_R0 Vn n - x + - (sum_f_R0 Wn n - x0)) <= Rabs (sum_f_R0 Vn n - x) + Rabs (- (sum_f_R0 Wn n - x0))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
Rabs (sum_f_R0 Vn n - x) + Rabs (- (sum_f_R0 Wn n - x0)) < eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
sum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat

Rabs (sum_f_R0 Vn n - x) + Rabs (- (sum_f_R0 Wn n - x0)) < eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
sum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat

Rabs (sum_f_R0 Vn n - x) + Rabs (sum_f_R0 Wn n - x0) < eps / 2 + eps / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
eps / 2 + eps / 2 <= eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
sum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat

Rabs (sum_f_R0 Vn n - x) < eps / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
Rabs (sum_f_R0 Wn n - x0) < eps / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
eps / 2 + eps / 2 <= eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
sum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat

Rabs (sum_f_R0 Wn n - x0) < eps / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
eps / 2 + eps / 2 <= eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
sum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat

eps / 2 + eps / 2 <= eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
sum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat

sum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> R
Wn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
i:nat

2 * An i = 2 * (/ 2 * (2 * Rabs (An i) + An i) - / 2 * (2 * Rabs (An i) - An i))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> R
Wn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
i:nat
2 <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> R
Wn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
i:nat

2 * An i = 1 * (2 * Rabs (An i) + An i) - 1 * (2 * Rabs (An i) - An i)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> R
Wn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
i:nat
2 <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> R
Wn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
i:nat
2 <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> R
Wn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
i:nat

2 <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> R
Wn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
i:nat
2 <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> R
Wn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0
x, x0, eps:R
H5:eps > 0
H6:0 < eps / 2
x1:nat
H8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2
x2:nat
H9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2
N:=Nat.max x1 x2:nat
n:nat
H7:(n >= N)%nat
i:nat

2 <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0
0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
x:R
p:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0
x0:R
p0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0
eps:R
H5:eps > 0

0 < eps / 2
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0

Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0

(forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)) -> Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)

(forall n : nat, / Wn n <= 2 * / Rabs (An n)) -> Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)

(forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)) -> Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0

0 < eps / 3 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Wn (S n) / Wn n)) 0 < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps / 3

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Wn (S n) / Wn n)) 0 < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat

R_dist (Rabs (Wn (S n) / Wn n)) 0 < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:R_dist (Rabs (An (S n) / An n)) 0 < eps / 3

R_dist (Rabs (Wn (S n) / Wn n)) 0 < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3

Wn (S n) / Wn n < eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3
Wn (S n) / Wn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3

Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3
3 * Rabs (An (S n) / An n) < eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3
Wn (S n) / Wn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3

3 * Rabs (An (S n) / An n) < eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3
Wn (S n) / Wn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3

0 < / 3
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3
/ 3 * (3 * Rabs (An (S n) / An n)) < / 3 * eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3
Wn (S n) / Wn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3

/ 3 * (3 * Rabs (An (S n) / An n)) < / 3 * eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3
Wn (S n) / Wn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3

Wn (S n) / Wn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3

0 < Wn (S n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3
0 < / Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
H6:forall n0 : nat, Wn (S n0) / Wn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H7:eps > 0
H8:0 < eps / 3
x:nat
H9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H10:(n >= x)%nat
H11:Rabs (An (S n) / An n) < eps / 3

0 < / Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
H6:forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
eps:R
H7:eps > 0

0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)
forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
H5:forall n : nat, / Wn n <= 2 * / Rabs (An n)

forall n : nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat

Wn (S n) * / Wn n <= Wn (S n) * 2 * / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
Wn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat

0 <= Wn (S n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
/ Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
Wn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat

/ Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
Wn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat

Wn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat

Wn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat

0 <= 2 * / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
Wn (S n) <= 3 * / 2 * Rabs (An (S n))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat

0 < 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
0 < / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
Wn (S n) <= 3 * / 2 * Rabs (An (S n))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat

0 < / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
Wn (S n) <= 3 * / 2 * Rabs (An (S n))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat

Wn (S n) <= 3 * / 2 * Rabs (An (S n))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat
An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
H5:forall n0 : nat, / Wn n0 <= 2 * / Rabs (An n0)
n:nat

An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
H4:forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)

forall n : nat, / Wn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

0 < Wn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Wn n * / Wn n <= Wn n * (2 * / Rabs (An n))
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

Wn n * / Wn n <= Wn n * (2 * / Rabs (An n))
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

1 <= Wn n * (2 * / Rabs (An n))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Wn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

0 < Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) * 1 <= Rabs (An n) * (Wn n * (2 * / Rabs (An n)))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Wn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

Rabs (An n) * 1 <= Rabs (An n) * (Wn n * (2 * / Rabs (An n)))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Wn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

Rabs (An n) <= 2 * Wn n * 1
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Wn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

0 < / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
/ 2 * Rabs (An n) <= / 2 * (2 * Wn n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Wn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

/ 2 * Rabs (An n) <= / 2 * (2 * Wn n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Wn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

/ 2 * Rabs (An n) <= 1 * Wn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
2 <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Wn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

2 <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Wn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Wn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

Wn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0

forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat

/ 2 * Rabs (An n) <= Wn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat
Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat

0 <= / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat
Rabs (An n) <= 2 * Rabs (An n) - An n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat
Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat

Rabs (An n) <= 2 * Rabs (An n) - An n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat
Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat

0 <= Rabs (An n) + - An n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat
Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat

An n + 0 <= An n + (Rabs (An n) + - An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat
Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat

Wn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat

0 <= / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat
2 * Rabs (An n) - An n <= 3 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat

2 * Rabs (An n) - An n <= 3 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0
n:nat

- An n <= Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n

Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n

(forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)) -> Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)

(forall n : nat, / Vn n <= 2 * / Rabs (An n)) -> Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)

(forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)) -> Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0

0 < eps / 3 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Vn (S n) / Vn n)) 0 < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps / 3

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Vn (S n) / Vn n)) 0 < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat

R_dist (Rabs (Vn (S n) / Vn n)) 0 < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:R_dist (Rabs (An (S n) / An n)) 0 < eps / 3

R_dist (Rabs (Vn (S n) / Vn n)) 0 < eps
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3

Vn (S n) / Vn n < eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3
Vn (S n) / Vn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3

Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3
3 * Rabs (An (S n) / An n) < eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3
Vn (S n) / Vn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3

3 * Rabs (An (S n) / An n) < eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3
Vn (S n) / Vn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3

0 < / 3
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3
/ 3 * (3 * Rabs (An (S n) / An n)) < / 3 * eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3
Vn (S n) / Vn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3

/ 3 * (3 * Rabs (An (S n) / An n)) < / 3 * eps
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3
Vn (S n) / Vn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3

Vn (S n) / Vn n >= 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3

0 < Vn (S n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3
0 < / Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
H5:forall n0 : nat, Vn (S n0) / Vn n0 <= 3 * Rabs (An (S n0) / An n0)
eps:R
H6:eps > 0
H7:0 < eps / 3
x:nat
H8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3
n:nat
H9:(n >= x)%nat
H10:Rabs (An (S n) / An n) < eps / 3

0 < / Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0
0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
H5:forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
eps:R
H6:eps > 0

0 < eps / 3
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)
forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
H4:forall n : nat, / Vn n <= 2 * / Rabs (An n)

forall n : nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat

Vn (S n) * / Vn n <= Vn (S n) * 2 * / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
Vn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat

0 <= Vn (S n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
/ Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
Vn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat

/ Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
Vn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat

Vn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat

Vn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat

0 <= 2 * / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
Vn (S n) <= 3 * / 2 * Rabs (An (S n))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat

0 < 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
0 < / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
Vn (S n) <= 3 * / 2 * Rabs (An (S n))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat

0 < / Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
Vn (S n) <= 3 * / 2 * Rabs (An (S n))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat

Vn (S n) <= 3 * / 2 * Rabs (An (S n))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat
An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
H4:forall n0 : nat, / Vn n0 <= 2 * / Rabs (An n0)
n:nat

An n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
H3:forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)

forall n : nat, / Vn n <= 2 * / Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Vn n * / Vn n <= Vn n * (2 * / Rabs (An n))
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

Vn n * / Vn n <= Vn n * (2 * / Rabs (An n))
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

1 <= Vn n * (2 * / Rabs (An n))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Vn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

0 < Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) * 1 <= Rabs (An n) * (Vn n * (2 * / Rabs (An n)))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Vn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

Rabs (An n) * 1 <= Rabs (An n) * (Vn n * (2 * / Rabs (An n)))
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Vn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

Rabs (An n) <= 2 * Vn n * 1
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Vn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

0 < / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
/ 2 * Rabs (An n) <= / 2 * (2 * Vn n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Vn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

/ 2 * Rabs (An n) <= / 2 * (2 * Vn n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Vn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

/ 2 * Rabs (An n) <= 1 * Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
2 <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Vn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

2 <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Vn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

Rabs (An n) <> 0
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat
Vn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)
n:nat

Vn n <> 0
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n
forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
H2:forall n : nat, 0 < Wn n

forall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat

/ 2 * Rabs (An n) <= Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat
Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat

0 <= / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat
Rabs (An n) <= 2 * Rabs (An n) + An n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat
Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat

Rabs (An n) <= 2 * Rabs (An n) + An n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat
Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat

0 <= Rabs (An n) + An n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat
Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat

Vn n <= 3 * / 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat

0 <= / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat
2 * Rabs (An n) + An n <= 3 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
H2:forall n0 : nat, 0 < Wn n0
n:nat

2 * Rabs (An n) + An n <= 3 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n
forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n : nat, 0 < Vn n

forall n : nat, 0 < Wn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
n:nat

0 < / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
n:nat
0 < 2 * Rabs (An n) - An n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
n:nat

0 < 2 * Rabs (An n) - An n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
n:nat

An n <= Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
n:nat
Rabs (An n) < 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
H1:forall n0 : nat, 0 < Vn n0
n:nat

Rabs (An n) < 2 * Rabs (An n)
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
forall n : nat, 0 < Vn n
An:nat -> R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R

forall n : nat, 0 < Vn n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
n:nat

0 < / 2
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
n:nat
0 < 2 * Rabs (An n) + An n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
n:nat

0 < 2 * Rabs (An n) + An n
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
n:nat

- An n <= Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
n:nat
Rabs (An n) < 2 * Rabs (An n)
An:nat -> R
H:forall n0 : nat, An n0 <> 0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0
Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> R
Wn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> R
n:nat

Rabs (An n) < 2 * Rabs (An n)
rewrite double; pattern (Rabs (An n)) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H. Defined.

forall (An : nat -> R) (x : R), x <> 0 -> (forall n : nat, An n <> 0) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0 -> {l : R | Pser An x l}

forall (An : nat -> R) (x : R), x <> 0 -> (forall n : nat, An n <> 0) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0 -> {l : R | Pser An x l}
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R

{l : R | Pser An x l}
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R

(forall n : nat, Bn n <> 0) -> {l : R | Pser An x l}
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0

Un_cv (fun n : nat => Rabs (Bn (S n) / Bn n)) 0 -> {l : R | Pser An x l}
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
Un_cv (fun n : nat => Rabs (Bn (S n) / Bn n)) 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
H3:Un_cv (fun n : nat => Rabs (Bn (S n) / Bn n)) 0
x0:R
H4:Un_cv (fun N : nat => sum_f_R0 Bn N) x0

{l : R | Pser An x l}
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
Un_cv (fun n : nat => Rabs (Bn (S n) / Bn n)) 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0

Un_cv (fun n : nat => Rabs (Bn (S n) / Bn n)) 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0

0 < eps / Rabs x -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Bn (S n) / Bn n)) 0 < eps
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps / Rabs x

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Bn (S n) / Bn n)) 0 < eps
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (An (S n) / An n * x) < eps
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

0 < / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
/ Rabs x * (Rabs (An (S n) / An n) * Rabs x) < / Rabs x * eps
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

/ Rabs x * (Rabs (An (S n) / An n) * Rabs x) < / Rabs x * eps
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

1 * Rabs (An (S n) / An n) < / Rabs x * eps
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) * / An n0)) 0 < eps * / Rabs x
n:nat
H6:(n >= x0)%nat

R_dist (Rabs (An (S n) * / An n)) 0 < eps * / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) * / An n0)) 0 < eps * / Rabs x
n:nat
H6:(n >= x0)%nat
R_dist (Rabs (An (S n) * / An n)) 0 = Rabs (An (S n) / An n)
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) * / An n0)) 0 < eps * / Rabs x
n:nat
H6:(n >= x0)%nat

R_dist (Rabs (An (S n) * / An n)) 0 = Rabs (An (S n) / An n)
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs x <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x ^ 1) * (/ An n * / x ^ n)
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * x ^ 1 * / An n * 1
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x ^ n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n0 : nat, Bn n0 <> 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x ^ n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0
Bn:=fun i : nat => An i * x ^ i:nat -> R
H2:forall n : nat, Bn n <> 0
eps:R
H3:eps > 0

0 < eps / Rabs x
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R
forall n : nat, Bn n <> 0
An:nat -> R
x:R
H:x <> 0
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Bn:=fun i : nat => An i * x ^ i:nat -> R

forall n : nat, Bn n <> 0
intro; unfold Bn; apply prod_neq_R0; [ apply H0 | apply pow_nonzero; assumption ]. Defined.

forall (An : nat -> R) (x : R), x = 0 -> {l : R | Pser An x l}

forall (An : nat -> R) (x : R), x = 0 -> {l : R | Pser An x l}
An:nat -> R
x:R
H:x = 0

Pser An x (An 0%nat)
An:nat -> R
x:R
H:x = 0
eps:R
H0:eps > 0
n:nat
H1:(n >= 0)%nat

R_dist (An 0%nat) (An 0%nat) < eps
An:nat -> R
x:R
H:x = 0
eps:R
H0:eps > 0
n:nat
H1:(n >= 0)%nat
An 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) n
An:nat -> R
x:R
H:x = 0
eps:R
H0:eps > 0
n:nat
H1:(n >= 0)%nat

An 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) n
An:nat -> R
x:R
H:x = 0
eps:R
H0:eps > 0
H1:(0 >= 0)%nat

An 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) 0
An:nat -> R
x:R
H:x = 0
eps:R
H0:eps > 0
n:nat
H1:(S n >= 0)%nat
Hrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) n
An 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) (S n)
An:nat -> R
x:R
H:x = 0
eps:R
H0:eps > 0
n:nat
H1:(S n >= 0)%nat
Hrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) n

An 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) (S n)
rewrite tech5; rewrite Hrecn; [ rewrite H; simpl; ring | unfold ge; apply le_O_n ]. Qed.
A useful criterion of convergence for power series

forall (An : nat -> R) (x : R), (forall n : nat, An n <> 0) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0 -> {l : R | Pser An x l}

forall (An : nat -> R) (x : R), (forall n : nat, An n <> 0) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0 -> {l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hlt:x < 0

{l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Heq:x = 0
{l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hgt:x > 0
{l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hlt:x < 0

x <> 0 -> {l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hlt:x < 0
x <> 0
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Heq:x = 0
{l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hgt:x > 0
{l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hlt:x < 0

x <> 0
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Heq:x = 0
{l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hgt:x > 0
{l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Heq:x = 0

{l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hgt:x > 0
{l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hgt:x > 0

{l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hgt:x > 0

x <> 0 -> {l : R | Pser An x l}
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hgt:x > 0
x <> 0
An:nat -> R
x:R
H:forall n : nat, An n <> 0
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0
Hgt:x > 0

x <> 0
red; intro; rewrite H1 in Hgt; elim (Rlt_irrefl _ Hgt). Defined.

forall (An : nat -> R) (k : R), 0 <= k < 1 -> (forall n : nat, 0 < An n) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) k -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

forall (An : nat -> R) (k : R), 0 <= k < 1 -> (forall n : nat, 0 < An n) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) k -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k

({l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}) -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

bound (EUn (fun N : nat => sum_f_R0 An N))
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)

bound (EUn (fun N : nat => sum_f_R0 An N))
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)

bound (EUn (fun N : nat => sum_f_R0 An N))
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x

bound (EUn (fun N : nat => sum_f_R0 An N))
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x

bound (EUn (fun N : nat => sum_f_R0 An N))
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x

is_upper_bound (EUn (fun N : nat => sum_f_R0 An N)) (sum_f_R0 An x0 + / (1 - x) * An (S x0))
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i

x1 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2

x1 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2

sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat

sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat

sum_f_R0 An x2 <= sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat

sum_f_R0 An x2 + 0 <= sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat

0 <= sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat

0 < sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
0 < / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat

forall n : nat, (n <= x0 - S x2)%nat -> 0 < An (S x2 + n)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
0 < / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat

0 < / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat

0 < / (1 - x)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
0 < An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat

0 < An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat
sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x2 < x0)%nat

sum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0

sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0

0 <= / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0

0 < / (1 - x)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0
0 < An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
H7:x1 = sum_f_R0 An x0

0 < An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= sum_f_R0 An x0 + / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0) -> sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0) <= / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0) <= / (1 - x) * An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

0 <= An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0) <= / (1 - x)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0) <= / (1 - x)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

(1 - x ^ S (x2 - S x0)) / (1 - x) <= / (1 - x)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

0 < 1 - x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
(1 - x) * ((1 - x ^ S (x2 - S x0)) * / (1 - x)) <= (1 - x) * / (1 - x)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

x < x + (1 - x)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
(1 - x) * ((1 - x ^ S (x2 - S x0)) * / (1 - x)) <= (1 - x) * / (1 - x)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

(1 - x) * ((1 - x ^ S (x2 - S x0)) * / (1 - x)) <= (1 - x) * / (1 - x)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

(1 - x ^ S (x2 - S x0)) * / (1 - x) * (1 - x) <= / (1 - x) * (1 - x)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

(1 - x ^ S (x2 - S x0)) * 1 <= 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
1 - x <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

x ^ S (x2 - S x0) + (1 - x ^ S (x2 - S x0)) <= x ^ S (x2 - S x0) + 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
1 - x <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

1 <= x ^ S (x2 - S x0) + 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
1 - x <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

0 <= x ^ S (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
1 - x <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

0 < x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
1 - x <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

0 <= k
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
k < x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
1 - x <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

k < x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
1 - x <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

1 - x <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

1 <> x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
H10:1 = x

False
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
H10:1 = x
H11:k < x
H12:x < 1

False
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)

x <> 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
H10:x = 1

False
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
H8:sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
H10:x = 1
H11:k < x
H12:x < 1

False
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0) * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= An (S x0 + 0)%nat * sum_f_R0 (fun i : nat => x ^ i) (x2 - S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

0 <= x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
forall i : nat, An (S x0 + S i)%nat < x * An (S x0 + i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

0 <= k
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
k < x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
forall i : nat, An (S x0 + S i)%nat < x * An (S x0 + i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

k < x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
forall i : nat, An (S x0 + S i)%nat < x * An (S x0 + i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

forall i : nat, An (S x0 + S i)%nat < x * An (S x0 + i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat

An (S x0 + S i)%nat < x * An (S x0 + i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat

(forall n : nat, (n >= x0)%nat -> An (S n) < x * An n) -> An (S x0 + S i)%nat < x * An (S x0 + i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
forall n : nat, (n >= x0)%nat -> An (S n) < x * An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
H9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An n

An (S x0 + S i)%nat < x * An (S x0 + i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
forall n : nat, (n >= x0)%nat -> An (S n) < x * An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
H9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An n

An (S (S x0 + i)) < x * An (S x0 + i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
H9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An n
S (S x0 + i) = (S x0 + S i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
forall n : nat, (n >= x0)%nat -> An (S n) < x * An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
H9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An n

(S x0 + i >= x0)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
H9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An n
S (S x0 + i) = (S x0 + S i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
forall n : nat, (n >= x0)%nat -> An (S n) < x * An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
H9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An n

(x0 <= S x0 + i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
H9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An n
S (S x0 + i) = (S x0 + S i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
forall n : nat, (n >= x0)%nat -> An (S n) < x * An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
H9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An n

S (S x0 + i) = (S x0 + S i)%nat
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat
forall n : nat, (n >= x0)%nat -> An (S n) < x * An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i:nat

forall n : nat, (n >= x0)%nat -> An (S n) < x * An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

An (S n) < x * An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

0 < / An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
/ An n * An (S n) < / An n * (x * An n)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

/ An n * An (S n) < / An n * (x * An n)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

An (S n) * / An n < x * An n * / An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

An (S n) * / An n < x * (An n * / An n)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

An (S n) * / An n < x * 1
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
An n <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

An (S n) * / An n < x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
An n <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

Rabs (An (S n) / An n) < x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
Rabs (An (S n) / An n) = An (S n) * / An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
An n <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

Rabs (An (S n) / An n) = An (S n) * / An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
An n <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

An (S n) / An n = An (S n) * / An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
An (S n) / An n >= 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
An n <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

An (S n) / An n >= 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
An n <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

0 < An (S n)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
0 < / An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
An n <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

0 < / An n
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
An n <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat

An n <> 0
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
H10:An n = 0

False
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n0 : nat, 0 < An n0
H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)
H3:k < x < 1
H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x0:nat
H5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < x
x1:R
H6:exists i0 : nat, x1 = sum_f_R0 An i0
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
i, n:nat
H8:(n >= x0)%nat
H10:An n = 0
H11:0 < An n

False
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

An (S x0 + 0)%nat = An (S x0)
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat
sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {l0 : R | Un_cv (fun N : nat => sum_f_R0 An N) l0}
H1:exists k0 : R, k < k0 < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)
x:R
H2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)
H3:k < x < 1
H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x
x0:nat
H5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < x
x1:R
H6:exists i : nat, x1 = sum_f_R0 An i
x2:nat
H7:x1 = sum_f_R0 An x2
l:(x0 < x2)%nat

sum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

exists x : R, EUn (fun N : nat => sum_f_R0 An N) x
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k

{l : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
Hyp:0 <= k < 1
H:forall n : nat, 0 < An n
H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
x:R
H1:is_lub (EUn (fun N : nat => sum_f_R0 An N)) x

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
exists x; apply Un_cv_crit_lub; [ unfold Un_growing; intro; rewrite tech5; pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; apply H | apply H1]. Qed.

forall (An : nat -> R) (k : R), 0 <= k < 1 -> (forall n : nat, An n <> 0) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) k -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

forall (An : nat -> R) (k : R), 0 <= k < 1 -> (forall n : nat, An n <> 0) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) k -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k

({l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}) -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

Cauchy_crit_series An
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

Cauchy_crit_series (fun i : nat => Rabs (An i))
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

({l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}) -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}

{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}

0 <= k < 1
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
forall n : nat, 0 < Rabs (An n)
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
Un_cv (fun n : nat => Rabs (Rabs (An (S n)) / Rabs (An n))) k
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}

forall n : nat, 0 < Rabs (An n)
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
Un_cv (fun n : nat => Rabs (Rabs (An (S n)) / Rabs (An n))) k
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}

Un_cv (fun n : nat => Rabs (Rabs (An (S n)) / Rabs (An n))) k
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}

forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Rabs (An (S n)) / Rabs (An n))) k < eps
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}

forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Rabs (An (S n)) / Rabs (An n))) k < eps
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}

forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Rabs (An (S n)) * / Rabs (An n))) k < eps
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
eps:R
H2:eps > 0

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Rabs (An (S n)) * / Rabs (An n))) k < eps
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
eps:R
H2:eps > 0
x:nat
H3:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Rabs (An (S n)) * / Rabs (An n))) k < eps
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps
n:nat
H4:(n >= x)%nat

R_dist (Rabs (Rabs (An (S n)) * / Rabs (An n))) k < eps
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps
n:nat
H4:(n >= x)%nat

R_dist (Rabs (Rabs (An (S n)) * Rabs (/ An n))) k < eps
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps
n:nat
H4:(n >= x)%nat
An n <> 0
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps
n:nat
H4:(n >= x)%nat

R_dist (Rabs (Rabs (An (S n) * / An n))) k < eps
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps
n:nat
H4:(n >= x)%nat
An n <> 0
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps
n:nat
H4:(n >= x)%nat

R_dist (Rabs (An (S n) * / An n)) k < eps
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps
n:nat
H4:(n >= x)%nat
An n <> 0
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Hyp:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps
n:nat
H4:(n >= x)%nat

An n <> 0
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
X:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}

{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
X:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
x:R
p:Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) x

{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
Hyp0:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
X:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) l}
x:R
p:Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) x

Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) x
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} -> {l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
x:R
p:Un_cv (fun N : nat => sum_f_R0 An N) x

{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
An:nat -> R
k:R
H:0 <= k < 1
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
X:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
x:R
p:Un_cv (fun N : nat => sum_f_R0 An N) x

Un_cv (fun N : nat => sum_f_R0 An N) x
assumption. Qed.
Convergence of power series in D(O,1/k) k=0 is described in Alembert_C3

forall (An : nat -> R) (x k : R), 0 < k -> (forall n : nat, An n <> 0) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) k -> Rabs x < / k -> {l : R | Pser An x l}

forall (An : nat -> R) (x k : R), 0 < k -> (forall n : nat, An n <> 0) -> Un_cv (fun n : nat => Rabs (An (S n) / An n)) k -> Rabs x < / k -> {l : R | Pser An x l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k

{l : R | Pser An x l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k

{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l} -> {l : R | Pser An x l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
X:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}

{l : R | Pser An x l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
X:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
x0:R
p:Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) x0

{l : R | Pser An x l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
X:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
x0:R
p:Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) x0

Pser An x x0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k

{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

0 <= k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

0 <= k * Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

0 <= k
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
0 <= Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

0 <= Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

0 < / k
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
/ k * (k * Rabs x) < / k * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

/ k * (k * Rabs x) < / k * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

/ k * k * Rabs x < / k * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

1 * Rabs x < / k * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
k <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

Rabs x < / k * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
k <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

k <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Hlt:x < 0
n:nat

An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Hlt:x < 0
n:nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Hlt:x < 0
n:nat

x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Hlt:x < 0
n:nat

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hlt:x < 0

Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps
H2:Rabs x < / k
Hlt:x < 0

forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0

0 < eps / Rabs x -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps / Rabs x

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps / Rabs x

forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

R_dist (Rabs (An (S n) / An n * x)) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs (An (S n) / An n * x) - k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs x * (Rabs (An (S n) / An n) - k)) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs x) * Rabs (Rabs (An (S n) / An n) - k) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs x * Rabs (Rabs (An (S n) / An n) - k) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

0 < / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

/ Rabs x * Rabs x * Rabs (Rabs (An (S n) / An n) - k) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

1 * Rabs (Rabs (An (S n) / An n) - k) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs (An (S n) / An n) - k) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs (An (S n) / An n) - k) < eps * / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> Rabs (Rabs (An (S n0) / An n0) - k) < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs (An (S n) / An n) - k) < eps * / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * x ^ (n + 1) * / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x ^ 1) * / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * (x * 1)) * / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x) * / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * / An n * x * (x ^ n * / x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * / An n * x * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0

0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0

0 < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0
0 < / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0

0 < / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hlt:x < 0
eps:R
H3:eps > 0

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0

{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0

Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) (An 0%nat)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0

forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 (fun i : nat => An i * x ^ i) n) (An 0%nat) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 (fun i : nat => An i * x ^ i) n) (An 0%nat) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0

forall n : nat, (n >= 0)%nat -> R_dist (sum_f_R0 (fun i : nat => An i * x ^ i) n) (An 0%nat) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(n >= 0)%nat

R_dist (sum_f_R0 (fun i : nat => An i * x ^ i) n) (An 0%nat) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(n >= 0)%nat

Rabs (sum_f_R0 (fun i : nat => An i * x ^ i) n - An 0%nat) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(n >= 0)%nat

Rabs (An 0%nat - An 0%nat) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(n >= 0)%nat
An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(n >= 0)%nat

An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
H4:(0 >= 0)%nat

An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(S n >= 0)%nat
Hrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n
An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) (S n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(S n >= 0)%nat
Hrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n

An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) (S n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(S n >= 0)%nat
Hrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n

An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n + An (S n) * x ^ S n
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(S n >= 0)%nat
Hrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n

An 0%nat = An 0%nat + An (S n) * x ^ S n
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(S n >= 0)%nat
Hrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n
(n >= 0)%nat
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Heq:x = 0
eps:R
H3:eps > 0
n:nat
H4:(S n >= 0)%nat
Hrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n

(n >= 0)%nat
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

0 <= k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

0 <= k * Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

0 <= k
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
0 <= Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

0 <= Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

k * Rabs x < 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

0 < / k
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
/ k * (k * Rabs x) < / k * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

/ k * (k * Rabs x) < / k * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

/ k * k * Rabs x < / k * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

1 * Rabs x < / k * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
k <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

Rabs x < / k * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
k <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

k <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

forall n : nat, An n * x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Hgt:x > 0
n:nat

An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Hgt:x > 0
n:nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Hgt:x > 0
n:nat

x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) k
H2:Rabs x < / k
Hgt:x > 0
n:nat

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0
Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) k
H2:Rabs x < / k
Hgt:x > 0

Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps
H2:Rabs x < / k
Hgt:x > 0

forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0

0 < eps / Rabs x -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps / Rabs x

exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps / Rabs x

forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

R_dist (Rabs (An (S n) / An n * x)) (k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs (An (S n) / An n * x) - k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs x * (Rabs (An (S n) / An n) - k)) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs x) * Rabs (Rabs (An (S n) / An n) - k) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs x * Rabs (Rabs (An (S n) / An n) - k) < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

0 < / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

/ Rabs x * Rabs x * Rabs (Rabs (An (S n) / An n) - k) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

1 * Rabs (Rabs (An (S n) / An n) - k) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs (An (S n) / An n) - k) < / Rabs x * eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs (An (S n) / An n) - k) < eps * / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> Rabs (Rabs (An (S n0) / An n0) - k) < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs (Rabs (An (S n) / An n) - k) < eps * / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
Rabs x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

Rabs x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * x ^ (n + 1) * / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x ^ 1) * / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * (x * 1)) * / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x) * / (An n * x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * / An n * x * (x ^ n * / x ^ n)
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An (n + 1)%nat * / An n * x = An (n + 1)%nat * / An n * x * 1
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

An n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat
x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x ^ n <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n0 : nat, An n0 <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
H4:0 < eps / Rabs x
x0:nat
H5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs x
n:nat
H6:(n >= x0)%nat

x <> 0
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0

0 < eps / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0

0 < eps
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0
0 < / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0

0 < / Rabs x
An:nat -> R
x, k:R
H:0 < k
H0:forall n : nat, An n <> 0
H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0
H2:Rabs x < / k
Hgt:x > 0
eps:R
H3:eps > 0

x <> 0
red; intro H7; rewrite H7 in Hgt; elim (Rlt_irrefl _ Hgt). Qed.