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(************************************************************************) (* * 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 -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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 -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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 -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2exists m : R, is_upper_bound (EUn (fun N : nat => sum_f_R0 An N)) mAn:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2exists m : R, is_upper_bound (EUn (fun N : nat => sum_f_R0 An N)) mAn:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2is_upper_bound (EUn (fun N : nat => sum_f_R0 An N)) (sum_f_R0 An x + 2 * An (S x))An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%natsum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) = sum_f_R0 An xAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%nat0 <= sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%natsum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) = sum_f_R0 An xAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%nat0 < sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%nat0 < 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%natsum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) = sum_f_R0 An xAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%nat0 < 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%natsum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) = sum_f_R0 An xAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x1 < x)%natsum_f_R0 An x1 + sum_f_R0 (fun i : nat => An (S x1 + i)%nat) (x - S x1) = sum_f_R0 An xAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2sum_f_R0 An x <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l : R | is_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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 20 <= 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x1 <= sum_f_R0 An x + 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) <= 2 * An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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) <= 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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) <= 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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) <= 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <= 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 - / 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <= 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 - / 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <= 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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) <= 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 - / 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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) <= 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 - / 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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) + 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 - / 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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) + 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 - / 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 - / 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 - / 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 - / 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 - 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natH3: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 <> 1An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%nat0 <= / 2An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natforall i : nat, An (S x + S i)%nat < / 2 * An (S x + i)%natAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natforall i : nat, An (S x + S i)%nat < / 2 * An (S x + i)%natAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%nati:nat(forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An n) -> An (S x + S i)%nat < / 2 * An (S x + i)%natAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%nati:natforall n : nat, (n >= x)%nat -> An (S n) < / 2 * An nAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%nati:natH4:forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An nAn (S (S x + i)) < / 2 * An (S x + i)%natAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%nati:natH4:forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An nS (S x + i) = (S x + S i)%natAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%nati:natforall n : nat, (n >= x)%nat -> An (S n) < / 2 * An nAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%nati:natH4:forall n : nat, (n >= x)%nat -> An (S n) < / 2 * An nS (S x + i) = (S x + S i)%natAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%nati:natforall n : nat, (n >= x)%nat -> An (S n) < / 2 * An nAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%nati:natforall n : nat, (n >= x)%nat -> An (S n) < / 2 * An nAn:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%nat0 < / An nAn:nat -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%nat/ An n * An (S n) < / An n * (/ 2 * An n)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%nat/ An n * An (S n) < / An n * (/ 2 * An n)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natAn (S n) * / An n < / 2 * 1An:nat -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natAn n <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natRabs (Rabs (An (S n) / An n) - 0) < / 2An:nat -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natRabs (Rabs (An (S n) / An n) - 0) = An (S n) * / An nAn:nat -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natAn n <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natRabs (Rabs (An (S n) / An n) - 0) = An (S n) * / An nAn:nat -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natAn n <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natAn (S n) / An n = An (S n) * / An nAn:nat -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natAn (S n) / An n >= 0An:nat -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natAn n <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natAn (S n) / An n >= 0An:nat -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natAn n <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n0 : nat, 0 < An n0H0:forall eps : R, eps > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n0 : nat, (n0 >= x)%nat -> Rabs (Rabs (An (S n0) / An n0) - 0) < / 2x1:natl:(x < x1)%nati, n:natH3:(n >= x)%natAn n <> 0An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natAn (S x + 0)%nat = An (S x)An:nat -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < epsX:{l0 : R | is_lub (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 < / 2x:natH2:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < / 2x1:natl:(x < x1)%natsum_f_R0 An x + sum_f_R0 (fun i : nat => An (S x + i)%nat) (x1 - S x) = sum_f_R0 An x1An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0X:{l : R | is_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) xAn:nat -> RH:forall n : nat, 0 < An nH0: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 -> RH:forall n : nat, 0 < An nH0: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}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.An:nat -> RH:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0x:RH1: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}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 -> RH:forall n : nat, An n <> 0H0: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:Un_cv (fun N : nat => sum_f_R0 Vn N) xx0:Rp0: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 An n) (x - x0) < epsAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n : nat, (n >= x1)%nat -> R_dist (sum_f_R0 Vn n) x < eps / 2exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 An n) (x - x0) < epsAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x, x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n : nat, (n >= x1)%nat -> R_dist (sum_f_R0 Vn n) x < eps / 2exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 An n) (x - x0) < epsAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x, x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n : nat, (n >= x1)%nat -> R_dist (sum_f_R0 Vn n) x < eps / 2x2:natH9:forall n : nat, (n >= x2)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps / 2exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 An n) (x - x0) < epsAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n : nat, (n >= x1)%nat -> R_dist (sum_f_R0 Vn n) x < eps / 2x2:natH9:forall n : nat, (n >= x2)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps / 2exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 An n) (x - x0) < epsAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n : nat, (n >= x1)%nat -> R_dist (sum_f_R0 Vn n) x < eps / 2x2:natH9:forall n : nat, (n >= x2)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps / 2N:=Nat.max x1 x2:natexists N0 : nat, forall n : nat, (n >= N0)%nat -> R_dist (sum_f_R0 An n) (x - x0) < epsAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natR_dist (sum_f_R0 Vn n - sum_f_R0 Wn n) (x - x0) < epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natsum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natRabs (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 -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natRabs (sum_f_R0 Vn n - x) + Rabs (- (sum_f_R0 Wn n - x0)) < epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natsum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natRabs (sum_f_R0 Vn n - x) + Rabs (- (sum_f_R0 Wn n - x0)) < epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natsum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natRabs (sum_f_R0 Vn n - x) + Rabs (sum_f_R0 Wn n - x0) < eps / 2 + eps / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nateps / 2 + eps / 2 <= epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natsum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natRabs (sum_f_R0 Vn n - x) < eps / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natRabs (sum_f_R0 Wn n - x0) < eps / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nateps / 2 + eps / 2 <= epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natsum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natRabs (sum_f_R0 Wn n - x0) < eps / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nateps / 2 + eps / 2 <= epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natsum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nateps / 2 + eps / 2 <= epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natsum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%natsum_f_R0 Vn n - sum_f_R0 Wn n = sum_f_R0 An nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> RWn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nati:nat2 * An i = 2 * (/ 2 * (2 * Rabs (An i) + An i) - / 2 * (2 * Rabs (An i) - An i))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> RWn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nati:nat2 <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> RWn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nati:nat2 * An i = 1 * (2 * Rabs (An i) + An i) - 1 * (2 * Rabs (An i) - An i)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> RWn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nati:nat2 <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> RWn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nati:nat2 <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> RWn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nati:nat2 <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> RWn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nati:nat2 <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i0 : nat => (2 * Rabs (An i0) + An i0) / 2:nat -> RWn:=fun i0 : nat => (2 * Rabs (An i0) - An i0) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:Un_cv (fun n0 : nat => Rabs (Wn (S n0) / Wn n0)) 0x, x0, eps:RH5:eps > 0H6:0 < eps / 2x1:natH8:forall n0 : nat, (n0 >= x1)%nat -> R_dist (sum_f_R0 Vn n0) x < eps / 2x2:natH9:forall n0 : nat, (n0 >= x2)%nat -> R_dist (sum_f_R0 Wn n0) x0 < eps / 2N:=Nat.max x1 x2:natn:natH7:(n >= N)%nati:nat2 <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4:Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0x:Rp:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Vn n) x < eps0x0:Rp0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 Wn n) x0 < eps0eps:RH5:eps > 00 < eps / 2An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0Un_cv (fun n : nat => Rabs (Wn (S n) / Wn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Wn (S n) / Wn n)) 0 < epsAn:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps / 3exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Wn (S n) / Wn n)) 0 < epsAn:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natR_dist (Rabs (Wn (S n) / Wn n)) 0 < epsAn:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:R_dist (Rabs (An (S n) / An n)) 0 < eps / 3R_dist (Rabs (Wn (S n) / Wn n)) 0 < epsAn:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 3Wn (S n) / Wn n < epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 3Wn (S n) / Wn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 3Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 33 * Rabs (An (S n) / An n) < epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 3Wn (S n) / Wn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 33 * Rabs (An (S n) / An n) < epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 3Wn (S n) / Wn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 30 < / 3An:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 3/ 3 * (3 * Rabs (An (S n) / An n)) < / 3 * epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 3Wn (S n) / Wn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 3/ 3 * (3 * Rabs (An (S n) / An n)) < / 3 * epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 3Wn (S n) / Wn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 3Wn (S n) / Wn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 30 < Wn (S n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 30 < / Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:RH7:eps > 0H8:0 < eps / 3x:natH9:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH10:(n >= x)%natH11:Rabs (An (S n) / An n) < eps / 30 < / Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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:RH7:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natWn (S n) * / Wn n <= Wn (S n) * 2 * / Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natWn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:nat0 <= Wn (S n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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 -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natWn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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 -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natWn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natWn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natWn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * / Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:nat0 <= 2 * / Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natWn (S n) <= 3 * / 2 * Rabs (An (S n))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:nat0 < 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:nat0 < / Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natWn (S n) <= 3 * / 2 * Rabs (An (S n))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:nat0 < / Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natWn (S n) <= 3 * / 2 * Rabs (An (S n))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natWn (S n) <= 3 * / 2 * Rabs (An (S n))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0H4: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:nat0 < Wn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n * / Wn n <= Wn n * (2 * / Rabs (An n))An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n * / Wn n <= Wn n * (2 * / Rabs (An n))An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:nat1 <= Wn n * (2 * / Rabs (An n))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:nat0 < Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) * 1 <= Rabs (An n) * (Wn n * (2 * / Rabs (An n)))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) * 1 <= Rabs (An n) * (Wn n * (2 * / Rabs (An n)))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <= 2 * Wn n * 1An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:nat0 < / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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 -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4: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 -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:nat/ 2 * Rabs (An n) <= 1 * Wn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:nat2 <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:nat2 <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0H4:forall n0 : nat, / 2 * Rabs (An n0) <= Wn n0 <= 3 * / 2 * Rabs (An n0)n:natWn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3:Un_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0forall n : nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:nat/ 2 * Rabs (An n) <= Wn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:natWn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:nat0 <= / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:natRabs (An n) <= 2 * Rabs (An n) - An nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:natWn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:natRabs (An n) <= 2 * Rabs (An n) - An nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:natWn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:nat0 <= Rabs (An n) + - An nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:natWn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:natAn n + 0 <= An n + (Rabs (An n) + - An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:natWn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:natWn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:nat0 <= / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:nat2 * Rabs (An n) - An n <= 3 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:nat2 * Rabs (An n) - An n <= 3 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:Un_cv (fun n0 : nat => Rabs (Vn (S n0) / Vn n0)) 0n:nat- An n <= Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nUn_cv (fun n : nat => Rabs (Vn (S n) / Vn n)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2: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)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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)) 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Vn (S n) / Vn n)) 0 < epsAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps / 3exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Vn (S n) / Vn n)) 0 < epsAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natR_dist (Rabs (Vn (S n) / Vn n)) 0 < epsAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:R_dist (Rabs (An (S n) / An n)) 0 < eps / 3R_dist (Rabs (Vn (S n) / Vn n)) 0 < epsAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 3Vn (S n) / Vn n < epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 3Vn (S n) / Vn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 3Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 33 * Rabs (An (S n) / An n) < epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 3Vn (S n) / Vn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 33 * Rabs (An (S n) / An n) < epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 3Vn (S n) / Vn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 30 < / 3An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 3/ 3 * (3 * Rabs (An (S n) / An n)) < / 3 * epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 3Vn (S n) / Vn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 3/ 3 * (3 * Rabs (An (S n) / An n)) < / 3 * epsAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 3Vn (S n) / Vn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 3Vn (S n) / Vn n >= 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 30 < Vn (S n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 30 < / Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:RH6:eps > 0H7:0 < eps / 3x:natH8:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / 3n:natH9:(n >= x)%natH10:Rabs (An (S n) / An n) < eps / 30 < / Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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:RH6:eps > 00 < eps / 3An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natVn (S n) * / Vn n <= Vn (S n) * 2 * / Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natVn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:nat0 <= Vn (S n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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 -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natVn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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 -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natVn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natVn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * Rabs (/ An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natVn (S n) * 2 * / Rabs (An n) <= 2 * (3 * / 2) * Rabs (An (S n)) * / Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:nat0 <= 2 * / Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natVn (S n) <= 3 * / 2 * Rabs (An (S n))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:nat0 < 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:nat0 < / Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natVn (S n) <= 3 * / 2 * Rabs (An (S n))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:nat0 < / Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natVn (S n) <= 3 * / 2 * Rabs (An (S n))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natVn (S n) <= 3 * / 2 * Rabs (An (S n))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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:natAn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nH3: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 -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:nat0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n * / Vn n <= Vn n * (2 * / Rabs (An n))An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n * / Vn n <= Vn n * (2 * / Rabs (An n))An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:nat1 <= Vn n * (2 * / Rabs (An n))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:nat0 < Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) * 1 <= Rabs (An n) * (Vn n * (2 * / Rabs (An n)))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) * 1 <= Rabs (An n) * (Vn n * (2 * / Rabs (An n)))An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <= 2 * Vn n * 1An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:nat0 < / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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 -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3: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 -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:nat/ 2 * Rabs (An n) <= 1 * Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:nat2 <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:nat2 <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natRabs (An n) <> 0An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0H3:forall n0 : nat, / 2 * Rabs (An n0) <= Vn n0 <= 3 * / 2 * Rabs (An n0)n:natVn n <> 0An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nH2:forall n : nat, 0 < Wn nforall n : nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:nat/ 2 * Rabs (An n) <= Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:natVn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:nat0 <= / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:natRabs (An n) <= 2 * Rabs (An n) + An nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:natVn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:natRabs (An n) <= 2 * Rabs (An n) + An nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:natVn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:nat0 <= Rabs (An n) + An nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:natVn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:natVn n <= 3 * / 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:nat0 <= / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:nat2 * Rabs (An n) + An n <= 3 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0H2:forall n0 : nat, 0 < Wn n0n:nat2 * Rabs (An n) + An n <= 3 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n : nat, 0 < Vn nforall n : nat, 0 < Wn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0n:nat0 < / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0n:nat0 < 2 * Rabs (An n) - An nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0n:nat0 < 2 * Rabs (An n) - An nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0n:natAn n <= Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0n:natRabs (An n) < 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> RH1:forall n0 : nat, 0 < Vn n0n:natRabs (An n) < 2 * Rabs (An n)An:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rforall n : nat, 0 < Vn nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rn:nat0 < / 2An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rn:nat0 < 2 * Rabs (An n) + An nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rn:nat0 < 2 * Rabs (An n) + An nAn:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rn:nat- An n <= Rabs (An n)An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rn:natRabs (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.An:nat -> RH:forall n0 : nat, An n0 <> 0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) 0Vn:=fun i : nat => (2 * Rabs (An i) + An i) / 2:nat -> RWn:=fun i : nat => (2 * Rabs (An i) - An i) / 2:nat -> Rn:natRabs (An n) < 2 * Rabs (An n)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 -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> R{l : R | Pser An x l}An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> R(forall n : nat, Bn n <> 0) -> {l : R | Pser An x l}An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0Un_cv (fun n : nat => Rabs (Bn (S n) / Bn n)) 0 -> {l : R | Pser An x l}An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0Un_cv (fun n : nat => Rabs (Bn (S n) / Bn n)) 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0H3:Un_cv (fun n : nat => Rabs (Bn (S n) / Bn n)) 0x0:RH4:Un_cv (fun N : nat => sum_f_R0 Bn N) x0{l : R | Pser An x l}An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0Un_cv (fun n : nat => Rabs (Bn (S n) / Bn n)) 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0Un_cv (fun n : nat => Rabs (Bn (S n) / Bn n)) 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs x -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Bn (S n) / Bn n)) 0 < epsAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps / Rabs xexists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Bn (S n) / Bn n)) 0 < epsAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natRabs (An (S n) / An n * x) < epsAn:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%nat0 < / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%nat/ Rabs x * (Rabs (An (S n) / An n) * Rabs x) < / Rabs x * epsAn:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%nat/ Rabs x * (Rabs (An (S n) / An n) * Rabs x) < / Rabs x * epsAn:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%nat1 * Rabs (An (S n) / An n) < / Rabs x * epsAn:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) * / An n0)) 0 < eps * / Rabs xn:natH6:(n >= x0)%natR_dist (Rabs (An (S n) * / An n)) 0 < eps * / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) * / An n0)) 0 < eps * / Rabs xn:natH6:(n >= x0)%natR_dist (Rabs (An (S n) * / An n)) 0 = Rabs (An (S n) / An n)An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) * / An n0)) 0 < eps * / Rabs xn:natH6:(n >= x0)%natR_dist (Rabs (An (S n) * / An n)) 0 = Rabs (An (S n) / An n)An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x ^ 1) * (/ An n * / x ^ n)An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * x ^ 1 * / An n * 1An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n0 : nat, Bn n0 <> 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) 0 < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) 0 < eps0Bn:=fun i : nat => An i * x ^ i:nat -> RH2:forall n : nat, Bn n <> 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0intro; unfold Bn; apply prod_neq_R0; [ apply H0 | apply pow_nonzero; assumption ]. Defined.An:nat -> Rx:RH:x <> 0H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Bn:=fun i : nat => An i * x ^ i:nat -> Rforall n : nat, Bn n <> 0forall (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 -> Rx:RH:x = 0Pser An x (An 0%nat)An:nat -> Rx:RH:x = 0eps:RH0:eps > 0n:natH1:(n >= 0)%natR_dist (An 0%nat) (An 0%nat) < epsAn:nat -> Rx:RH:x = 0eps:RH0:eps > 0n:natH1:(n >= 0)%natAn 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) nAn:nat -> Rx:RH:x = 0eps:RH0:eps > 0n:natH1:(n >= 0)%natAn 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) nAn:nat -> Rx:RH:x = 0eps:RH0:eps > 0H1:(0 >= 0)%natAn 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) 0An:nat -> Rx:RH:x = 0eps:RH0:eps > 0n:natH1:(S n >= 0)%natHrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) nAn 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.An:nat -> Rx:RH:x = 0eps:RH0:eps > 0n:natH1:(S n >= 0)%natHrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) nAn 0%nat = sum_f_R0 (fun n0 : nat => An n0 * x ^ n0) (S n)
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 -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hlt:x < 0{l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Heq:x = 0{l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hgt:x > 0{l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hlt:x < 0x <> 0 -> {l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hlt:x < 0x <> 0An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Heq:x = 0{l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hgt:x > 0{l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hlt:x < 0x <> 0An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Heq:x = 0{l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hgt:x > 0{l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Heq:x = 0{l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hgt:x > 0{l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hgt:x > 0{l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hgt:x > 0x <> 0 -> {l : R | Pser An x l}An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hgt:x > 0x <> 0red; intro; rewrite H1 in Hgt; elim (Rlt_irrefl _ Hgt). Defined.An:nat -> Rx:RH:forall n : nat, An n <> 0H0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) 0Hgt:x > 0x <> 0forall (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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xbound (EUn (fun N : nat => sum_f_R0 An N))An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xbound (EUn (fun N : nat => sum_f_R0 An N))An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xis_upper_bound (EUn (fun N : nat => sum_f_R0 An N)) (sum_f_R0 An x0 + / (1 - x) * An (S x0))An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix1 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2x1 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2sum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%nat0 <= sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%nat0 < sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%nat0 < / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natforall n : nat, (n <= x0 - S x2)%nat -> 0 < An (S x2 + n)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%nat0 < / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%nat0 < / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%nat0 < / (1 - x)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%nat0 < An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%nat0 < An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x2 < x0)%natsum_f_R0 An x2 + sum_f_R0 (fun i : nat => An (S x2 + i)%nat) (x0 - S x2) = sum_f_R0 An x0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x0sum_f_R0 An x0 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x00 <= / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x00 < / (1 - x)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x00 < An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An iH7:x1 = sum_f_R0 An x00 < An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x2 <= sum_f_R0 An x0 + / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) <= / (1 - x) * An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 - xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <= 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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) + 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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) + 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 < xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <= kAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 < xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 < xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 = xFalseAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 = xH11:k < xH12:x < 1FalseAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 <> 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 = 1FalseAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natH8: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 = 1H11:k < xH12:x < 1FalseAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nat0 <= xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natforall i : nat, An (S x0 + S i)%nat < x * An (S x0 + i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nat0 <= kAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natk < xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natforall i : nat, An (S x0 + S i)%nat < x * An (S x0 + i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natk < xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natforall i : nat, An (S x0 + S i)%nat < x * An (S x0 + i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natforall i : nat, An (S x0 + S i)%nat < x * An (S x0 + i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natAn (S x0 + S i)%nat < x * An (S x0 + i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:nat(forall n : nat, (n >= x0)%nat -> An (S n) < x * An n) -> An (S x0 + S i)%nat < x * An (S x0 + i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natforall n : nat, (n >= x0)%nat -> An (S n) < x * An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natH9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An nAn (S x0 + S i)%nat < x * An (S x0 + i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natforall n : nat, (n >= x0)%nat -> An (S n) < x * An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natH9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An nAn (S (S x0 + i)) < x * An (S x0 + i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natH9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An nS (S x0 + i) = (S x0 + S i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natforall n : nat, (n >= x0)%nat -> An (S n) < x * An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natH9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An n(S x0 + i >= x0)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natH9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An nS (S x0 + i) = (S x0 + S i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natforall n : nat, (n >= x0)%nat -> An (S n) < x * An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natH9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An n(x0 <= S x0 + i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natH9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An nS (S x0 + i) = (S x0 + S i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natforall n : nat, (n >= x0)%nat -> An (S n) < x * An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natH9:forall n : nat, (n >= x0)%nat -> An (S n) < x * An nS (S x0 + i) = (S x0 + S i)%natAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natforall n : nat, (n >= x0)%nat -> An (S n) < x * An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati:natforall n : nat, (n >= x0)%nat -> An (S n) < x * An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn (S n) < x * An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%nat0 < / An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%nat/ An n * An (S n) < / An n * (x * An n)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%nat/ An n * An (S n) < / An n * (x * An n)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn (S n) * / An n < x * An n * / An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn (S n) * / An n < x * (An n * / An n)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn (S n) * / An n < x * 1An:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn n <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn (S n) * / An n < xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn n <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natRabs (An (S n) / An n) < xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natRabs (An (S n) / An n) = An (S n) * / An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn n <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natRabs (An (S n) / An n) = An (S n) * / An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn n <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn (S n) / An n = An (S n) * / An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn (S n) / An n >= 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn n <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn (S n) / An n >= 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn n <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%nat0 < An (S n)An:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%nat0 < / An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn n <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%nat0 < / An nAn:nat -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn n <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natAn n <> 0An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natH10:An n = 0FalseAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n0 : nat, 0 < An n0H0:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < x)H3:k < x < 1H4:exists N : nat, forall n0 : nat, (N <= n0)%nat -> Rabs (An (S n0) / An n0) < xx0:natH5:forall n0 : nat, (x0 <= n0)%nat -> Rabs (An (S n0) / An n0) < xx1:RH6:exists i0 : nat, x1 = sum_f_R0 An i0x2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%nati, n:natH8:(n >= x0)%natH10:An n = 0H11:0 < An nFalseAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natAn (S x0 + 0)%nat = An (S x0)An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l0 : R | is_lub (EUn (fun N : nat => sum_f_R0 An N)) l0} -> {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:RH2:k < x < 1 /\ (exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < x)H3:k < x < 1H4:exists N : nat, forall n : nat, (N <= n)%nat -> Rabs (An (S n) / An n) < xx0:natH5:forall n : nat, (x0 <= n)%nat -> Rabs (An (S n) / An n) < xx1:RH6:exists i : nat, x1 = sum_f_R0 An ix2:natH7:x1 = sum_f_R0 An x2l:(x0 < x2)%natsum_f_R0 An x0 + sum_f_R0 (fun i : nat => An (S x0 + i)%nat) (x2 - S x0) = sum_f_R0 An x2An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | is_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) xAn:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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 -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0: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}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.An:nat -> Rk:RHyp:0 <= k < 1H:forall n : nat, 0 < An nH0:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kx:RH1: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}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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 AnAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 < 1An:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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))) kAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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))) kAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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))) kAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 < epsAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < epsHyp0:{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 < epsAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < epsHyp0:{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 < epsAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0Hyp0:{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:RH2:eps > 0exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Rabs (An (S n)) * / Rabs (An n))) k < epsAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0Hyp0:{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:RH2:eps > 0x:natH3:forall n : nat, (n >= x)%nat -> R_dist (Rabs (An (S n) / An n)) k < epsexists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (Rabs (An (S n)) * / Rabs (An n))) k < epsAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0Hyp0:{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:RH2:eps > 0x:natH3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < epsn:natH4:(n >= x)%natR_dist (Rabs (Rabs (An (S n)) * / Rabs (An n))) k < epsAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0Hyp0:{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:RH2:eps > 0x:natH3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < epsn:natH4:(n >= x)%natR_dist (Rabs (Rabs (An (S n)) * Rabs (/ An n))) k < epsAn:nat -> Rk:RH:0 <= k < 1H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0Hyp0:{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:RH2:eps > 0x:natH3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < epsn:natH4:(n >= x)%natAn n <> 0An:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0Hyp0:{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:RH2:eps > 0x:natH3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < epsn:natH4:(n >= x)%natR_dist (Rabs (Rabs (An (S n) * / An n))) k < epsAn:nat -> Rk:RH:0 <= k < 1H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0Hyp0:{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:RH2:eps > 0x:natH3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < epsn:natH4:(n >= x)%natAn n <> 0An:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0Hyp0:{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:RH2:eps > 0x:natH3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < epsn:natH4:(n >= x)%natR_dist (Rabs (An (S n) * / An n)) k < epsAn:nat -> Rk:RH:0 <= k < 1H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0Hyp0:{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:RH2:eps > 0x:natH3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < epsn:natH4:(n >= x)%natAn n <> 0An:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0Hyp0:{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:RH2:eps > 0x:natH3:forall n0 : nat, (n0 >= x)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < epsn:natH4:(n >= x)%natAn n <> 0An:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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:Rp: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kHyp0:{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:Rp:Un_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) xUn_cv (fun N : nat => sum_f_R0 (fun i : nat => Rabs (An i)) N) xAn:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1: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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{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 -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}x:Rp:Un_cv (fun N : nat => sum_f_R0 An N) x{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}assumption. Qed.An:nat -> Rk:RH:0 <= k < 1H0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kX:{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}x:Rp:Un_cv (fun N : nat => sum_f_R0 An N) xUn_cv (fun N : nat => sum_f_R0 An N) x
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 -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / k{l : R | Pser An x l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2: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 -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / k{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kX:{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 -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / k{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kX:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}x0:Rp: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 -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / k{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kX:{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}x0:Rp:Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) x0Pser An x x0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / k{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / k{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 00 <= k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 00 <= k * Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 00 <= kAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 00 <= Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 00 <= Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 00 < / kAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0/ k * (k * Rabs x) < / k * 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0/ k * (k * Rabs x) < / k * 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0/ k * k * Rabs x < / k * 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 01 * Rabs x < / k * 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0k <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Rabs x < / k * 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0k <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0k <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHlt:x < 0n:natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHlt:x < 0n:natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHlt:x < 0n:natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHlt:x < 0n:natx <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHlt:x < 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < epsH2:Rabs x < / kHlt:x < 0forall 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) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < 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) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xexists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps / Rabs xexists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps / Rabs xforall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natR_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natR_dist (Rabs (An (S n) / An n * x)) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs (An (S n) / An n * x) - k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs x * (Rabs (An (S n) / An n) - k)) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs x) * Rabs (Rabs (An (S n) / An n) - k) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x * Rabs (Rabs (An (S n) / An n) - k) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat0 < / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat/ Rabs x * Rabs x * Rabs (Rabs (An (S n) / An n) - k) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat1 * Rabs (Rabs (An (S n) / An n) - k) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs (An (S n) / An n) - k) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs (An (S n) / An n) - k) < eps * / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> Rabs (Rabs (An (S n0) / An n0) - k) < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs (An (S n) / An n) - k) < eps * / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * x ^ (n + 1) * / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x ^ 1) * / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * (x * 1)) * / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x) * / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * / An n * x * (x ^ n * / x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * / An n * x * 1An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 00 < / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHlt:x < 0eps:RH3:eps > 0x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) (An 0%nat)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0forall 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) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (sum_f_R0 (fun i : nat => An i * x ^ i) n) (An 0%nat) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0forall n : nat, (n >= 0)%nat -> R_dist (sum_f_R0 (fun i : nat => An i * x ^ i) n) (An 0%nat) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(n >= 0)%natR_dist (sum_f_R0 (fun i : nat => An i * x ^ i) n) (An 0%nat) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(n >= 0)%natRabs (sum_f_R0 (fun i : nat => An i * x ^ i) n - An 0%nat) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(n >= 0)%natRabs (An 0%nat - An 0%nat) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(n >= 0)%natAn 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) nAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(n >= 0)%natAn 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) nAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0H4:(0 >= 0)%natAn 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(S n >= 0)%natHrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) nAn 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) (S n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(S n >= 0)%natHrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) nAn 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) (S n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(S n >= 0)%natHrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) nAn 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n + An (S n) * x ^ S nAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(S n >= 0)%natHrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) nAn 0%nat = An 0%nat + An (S n) * x ^ S nAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(S n >= 0)%natHrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n(n >= 0)%natAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHeq:x = 0eps:RH3:eps > 0n:natH4:(S n >= 0)%natHrecn:(n >= 0)%nat -> An 0%nat = sum_f_R0 (fun i : nat => An i * x ^ i) n(n >= 0)%natAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0{l : R | Un_cv (fun N : nat => sum_f_R0 (fun i : nat => An i * x ^ i) N) l}An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 00 <= k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 00 <= k * Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 00 <= kAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 00 <= Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 00 <= Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0k * Rabs x < 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 00 < / kAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0/ k * (k * Rabs x) < / k * 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0/ k * (k * Rabs x) < / k * 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0/ k * k * Rabs x < / k * 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 01 * Rabs x < / k * 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0k <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Rabs x < / k * 1An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0k <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0k <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0forall n : nat, An n * x ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHgt:x > 0n:natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHgt:x > 0n:natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHgt:x > 0n:natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:Un_cv (fun n0 : nat => Rabs (An (S n0) / An n0)) kH2:Rabs x < / kHgt:x > 0n:natx <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:Un_cv (fun n : nat => Rabs (An (S n) / An n)) kH2:Rabs x < / kHgt:x > 0Un_cv (fun n : nat => Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps : R, eps > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < epsH2:Rabs x < / kHgt:x > 0forall 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) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < 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) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xexists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps / Rabs xexists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps / Rabs xforall n : nat, (n >= x0)%nat -> R_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natR_dist (Rabs (An (S n) * x ^ S n / (An n * x ^ n))) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natR_dist (Rabs (An (S n) / An n * x)) (k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs (An (S n) / An n * x) - k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs x * (Rabs (An (S n) / An n) - k)) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs x) * Rabs (Rabs (An (S n) / An n) - k) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x * Rabs (Rabs (An (S n) / An n) - k) < epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat0 < / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat/ Rabs x * (Rabs x * Rabs (Rabs (An (S n) / An n) - k)) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat/ Rabs x * Rabs x * Rabs (Rabs (An (S n) / An n) - k) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%nat1 * Rabs (Rabs (An (S n) / An n) - k) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs (An (S n) / An n) - k) < / Rabs x * epsAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs (An (S n) / An n) - k) < eps * / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> Rabs (Rabs (An (S n0) / An n0) - k) < eps / Rabs xn:natH6:(n >= x0)%natRabs (Rabs (An (S n) / An n) - k) < eps * / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natRabs x <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (S n) / An n * x = An (S n) * x ^ S n / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * x ^ (n + 1) * / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x ^ 1) * / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * (x * 1)) * / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x) * / (An n * x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * / An n * x * (x ^ n * / x ^ n)An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn (n + 1)%nat * / An n * x = An (n + 1)%nat * / An n * x * 1An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natAn n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx ^ n <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n0 : nat, An n0 <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n0 : nat, (n0 >= N)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0H4:0 < eps / Rabs xx0:natH5:forall n0 : nat, (n0 >= x0)%nat -> R_dist (Rabs (An (S n0) / An n0)) k < eps / Rabs xn:natH6:(n >= x0)%natx <> 0An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < eps / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < epsAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < / Rabs xAn:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 00 < / Rabs xred; intro H7; rewrite H7 in Hgt; elim (Rlt_irrefl _ Hgt). Qed.An:nat -> Rx, k:RH:0 < kH0:forall n : nat, An n <> 0H1:forall eps0 : R, eps0 > 0 -> exists N : nat, forall n : nat, (n >= N)%nat -> R_dist (Rabs (An (S n) / An n)) k < eps0H2:Rabs x < / kHgt:x > 0eps:RH3:eps > 0x <> 0