PartSum.v

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(*  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)         *)
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Require Import Rbase.
Require Import Rfunctions.
Require Import Rseries.
Require Import Rcomplete.
Require Import Max.
Local Open Scope R_scope.

Lemma tech1 :
  forall (An:nat -> R) (N:nat),
    (forall n:nat, (n <= N)%nat -> 0 < An n) -> 0 < sum_f_R0 An N.forall (An : nat -> R) (N : nat),
(forall n : nat, (n <= N)%nat -> 0 < An n) ->
0 < sum_f_R0 An N
Proof.forall (An : nat -> R) (N : nat),
(forall n : nat, (n <= N)%nat -> 0 < An n) ->
0 < sum_f_R0 An N
  intros; induction  N as [| N HrecN].An:nat -> R
H:forall n : nat, (n <= 0)%nat -> 0 < An n
0 < sum_f_R0 An 0
An:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> 0 < An n
HrecN:(forall n : nat, (n <= N)%nat -> 0 < An n) ->
0 < sum_f_R0 An N
0 < sum_f_R0 An (S N)
  simpl; apply H; apply le_n.An:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> 0 < An n
HrecN:(forall n : nat, (n <= N)%nat -> 0 < An n) ->
0 < sum_f_R0 An N
0 < sum_f_R0 An (S N)
  simpl; apply Rplus_lt_0_compat.An:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> 0 < An n
HrecN:(forall n : nat, (n <= N)%nat -> 0 < An n) ->
0 < sum_f_R0 An N
0 < sum_f_R0 An N
An:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> 0 < An n
HrecN:(forall n : nat, (n <= N)%nat -> 0 < An n) ->
0 < sum_f_R0 An N
0 < An (S N)
  apply HrecN; intros; apply H; apply le_S; assumption.An:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> 0 < An n
HrecN:(forall n : nat, (n <= N)%nat -> 0 < An n) ->
0 < sum_f_R0 An N
0 < An (S N)
  apply H; apply le_n.
Qed.

(* Chasles' relation *)
Lemma tech2 :
  forall (An:nat -> R) (m n:nat),
    (m < n)%nat ->
    sum_f_R0 An n =
    sum_f_R0 An m + sum_f_R0 (fun i:nat => An (S m + i)%nat) (n - S m).forall (An : nat -> R) (m n : nat),
(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m)
Proof.forall (An : nat -> R) (m n : nat),
(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m)
  intros; induction  n as [| n Hrecn].An:nat -> R
m:nat
H:(m < 0)%nat
sum_f_R0 An 0 =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (0 - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
  elim (lt_n_O _ H).An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
  cut ((m < n)%nat \/ m = n).An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n ->
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  intro; elim H0; intro.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  replace (sum_f_R0 An (S n)) with (sum_f_R0 An n + An (S n));
  [ idtac | reflexivity ].An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
sum_f_R0 An n + An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  replace (S n - S m)%nat with (S (n - S m)).An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
sum_f_R0 An n + An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (S (n - S m))
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S (n - S m) = (S n - S m)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  replace (sum_f_R0 (fun i:nat => An (S m + i)%nat) (S (n - S m))) with
  (sum_f_R0 (fun i:nat => An (S m + i)%nat) (n - S m) +
    An (S m + S (n - S m))%nat); [ idtac | reflexivity ].An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
sum_f_R0 An n + An (S n) =
sum_f_R0 An m +
(sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) +
 An (S m + S (n - S m))%nat)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S (n - S m) = (S n - S m)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  replace (S m + S (n - S m))%nat with (S n).An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
sum_f_R0 An n + An (S n) =
sum_f_R0 An m +
(sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) +
 An (S n))
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S n = (S m + S (n - S m))%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S (n - S m) = (S n - S m)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  rewrite (Hrecn H1).An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) +
An (S n) =
sum_f_R0 An m +
(sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) +
 An (S n))
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S n = (S m + S (n - S m))%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S (n - S m) = (S n - S m)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  ring.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S n = (S m + S (n - S m))%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S (n - S m) = (S n - S m)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  apply INR_eq; rewrite S_INR; rewrite plus_INR; do 2 rewrite S_INR;
    rewrite minus_INR.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
INR n + 1 = INR m + 1 + (INR n - INR (S m) + 1)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
(S m <= n)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S (n - S m) = (S n - S m)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  rewrite S_INR; ring.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
(S m <= n)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S (n - S m) = (S n - S m)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  apply lt_le_S; assumption.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
S (n - S m) = (S n - S m)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  apply INR_eq; rewrite S_INR; repeat rewrite minus_INR.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
INR n - INR (S m) + 1 = INR (S n) - INR (S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
(S m <= S n)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
(S m <= n)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  repeat rewrite S_INR; ring.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
(S m <= S n)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
(S m <= n)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  apply le_n_S; apply lt_le_weak; assumption.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:(m < n)%nat
(S m <= n)%nat
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  apply lt_le_S; assumption.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An (S n) =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat) (S n - S m)
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  rewrite H1; rewrite <- minus_n_n; simpl.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H0:(m < n)%nat \/ m = n
H1:m = n
sum_f_R0 An n + An (S n) =
sum_f_R0 An n + An (S (n + 0))
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  replace (n + 0)%nat with n; [ reflexivity | ring ].An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
(m < n)%nat \/ m = n
  inversion H.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
H1:m = n
(n < n)%nat \/ n = n
An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
m0:nat
H1:(S m <= n)%nat
H0:m0 = n
(m < n)%nat \/ m = n
  right; reflexivity.An:nat -> R
m, n:nat
H:(m < S n)%nat
Hrecn:(m < n)%nat ->
sum_f_R0 An n =
sum_f_R0 An m +
sum_f_R0 (fun i : nat => An (S m + i)%nat)
  (n - S m)
m0:nat
H1:(S m <= n)%nat
H0:m0 = n
(m < n)%nat \/ m = n
  left; apply lt_le_trans with (S m); [ apply lt_n_Sn | assumption ].
Qed.

(* Sum of geometric sequences *)
Lemma tech3 :
  forall (k:R) (N:nat),
    k <> 1 -> sum_f_R0 (fun i:nat => k ^ i) N = (1 - k ^ S N) / (1 - k).forall (k : R) (N : nat),
k <> 1 ->
sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
Proof.forall (k : R) (N : nat),
k <> 1 ->
sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
  intros; cut (1 - k <> 0).k:R
N:nat
H:k <> 1
1 - k <> 0 ->
sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
k:R
N:nat
H:k <> 1
1 - k <> 0
  intro; induction  N as [| N HrecN].k:R
H:k <> 1
H0:1 - k <> 0
sum_f_R0 (fun i : nat => k ^ i) 0 =
(1 - k ^ 1) / (1 - k)
k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
sum_f_R0 (fun i : nat => k ^ i) (S N) =
(1 - k ^ S (S N)) / (1 - k)
k:R
N:nat
H:k <> 1
1 - k <> 0
  simpl; rewrite Rmult_1_r; unfold Rdiv; rewrite <- Rinv_r_sym.k:R
H:k <> 1
H0:1 - k <> 0
1 = 1
k:R
H:k <> 1
H0:1 - k <> 0
1 - k <> 0
k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
sum_f_R0 (fun i : nat => k ^ i) (S N) =
(1 - k ^ S (S N)) / (1 - k)
k:R
N:nat
H:k <> 1
1 - k <> 0
  reflexivity.k:R
H:k <> 1
H0:1 - k <> 0
1 - k <> 0
k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
sum_f_R0 (fun i : nat => k ^ i) (S N) =
(1 - k ^ S (S N)) / (1 - k)
k:R
N:nat
H:k <> 1
1 - k <> 0
  apply H0.k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
sum_f_R0 (fun i : nat => k ^ i) (S N) =
(1 - k ^ S (S N)) / (1 - k)
k:R
N:nat
H:k <> 1
1 - k <> 0
  replace (sum_f_R0 (fun i:nat => k ^ i) (S N)) with
  (sum_f_R0 (fun i:nat => k ^ i) N + k ^ S N); [ idtac | reflexivity ];
  rewrite HrecN;
    replace ((1 - k ^ S N) / (1 - k) + k ^ S N) with
    ((1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k)).k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
(1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k) =
(1 - k ^ S (S N)) / (1 - k)
k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
(1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k) =
(1 - k ^ S N) / (1 - k) + k ^ S N
k:R
N:nat
H:k <> 1
1 - k <> 0
  apply Rmult_eq_reg_l with (1 - k).k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
(1 - k) *
((1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k)) =
(1 - k) * ((1 - k ^ S (S N)) / (1 - k))
k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
1 - k <> 0
k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
(1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k) =
(1 - k ^ S N) / (1 - k) + k ^ S N
k:R
N:nat
H:k <> 1
1 - k <> 0
  unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ (1 - k)));
    repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
      [ do 2 rewrite Rmult_1_l; simpl; ring | apply H0 ].k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
1 - k <> 0
k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
(1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k) =
(1 - k ^ S N) / (1 - k) + k ^ S N
k:R
N:nat
H:k <> 1
1 - k <> 0
  apply H0.k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
(1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k) =
(1 - k ^ S N) / (1 - k) + k ^ S N
k:R
N:nat
H:k <> 1
1 - k <> 0
  unfold Rdiv; rewrite Rmult_plus_distr_r; rewrite (Rmult_comm (1 - k));
    repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
(1 - k ^ S N) * / (1 - k) + k ^ S N * 1 =
(1 - k ^ S N) * / (1 - k) + k ^ S N
k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
1 - k <> 0
k:R
N:nat
H:k <> 1
1 - k <> 0
  rewrite Rmult_1_r; reflexivity.k:R
N:nat
H:k <> 1
H0:1 - k <> 0
HrecN:sum_f_R0 (fun i : nat => k ^ i) N =
(1 - k ^ S N) / (1 - k)
1 - k <> 0
k:R
N:nat
H:k <> 1
1 - k <> 0
  apply H0.k:R
N:nat
H:k <> 1
1 - k <> 0
  apply Rminus_eq_contra; red; intro; elim H; symmetry ;
    assumption.
Qed.

Lemma tech4 :
  forall (An:nat -> R) (k:R) (N:nat),
    0 <= k -> (forall i:nat, An (S i) < k * An i) -> An N <= An 0%nat * k ^ N.forall (An : nat -> R) (k : R) (N : nat),
0 <= k ->
(forall i : nat, An (S i) < k * An i) ->
An N <= An 0%nat * k ^ N
Proof.forall (An : nat -> R) (k : R) (N : nat),
0 <= k ->
(forall i : nat, An (S i) < k * An i) ->
An N <= An 0%nat * k ^ N
  intros; induction  N as [| N HrecN].An:nat -> R
k:R
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
An 0%nat <= An 0%nat * k ^ 0
An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:An N <= An 0%nat * k ^ N
An (S N) <= An 0%nat * k ^ S N
  simpl; right; ring.An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:An N <= An 0%nat * k ^ N
An (S N) <= An 0%nat * k ^ S N
  apply Rle_trans with (k * An N).An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:An N <= An 0%nat * k ^ N
An (S N) <= k * An N
An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:An N <= An 0%nat * k ^ N
k * An N <= An 0%nat * k ^ S N
  left; apply (H0 N).An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:An N <= An 0%nat * k ^ N
k * An N <= An 0%nat * k ^ S N
  replace (S N) with (N + 1)%nat; [ idtac | ring ].An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:An N <= An 0%nat * k ^ N
k * An N <= An 0%nat * k ^ (N + 1)
  rewrite pow_add; simpl; rewrite Rmult_1_r;
    replace (An 0%nat * (k ^ N * k)) with (k * (An 0%nat * k ^ N));
    [ idtac | ring ]; apply Rmult_le_compat_l.An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:An N <= An 0%nat * k ^ N
0 <= k
An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:An N <= An 0%nat * k ^ N
An N <= An 0%nat * k ^ N
  assumption.An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:An N <= An 0%nat * k ^ N
An N <= An 0%nat * k ^ N
  apply HrecN.
Qed.

Lemma tech5 :
  forall (An:nat -> R) (N:nat), sum_f_R0 An (S N) = sum_f_R0 An N + An (S N).forall (An : nat -> R) (N : nat),
sum_f_R0 An (S N) = sum_f_R0 An N + An (S N)
Proof.forall (An : nat -> R) (N : nat),
sum_f_R0 An (S N) = sum_f_R0 An N + An (S N)
  intros; reflexivity.
Qed.

Lemma tech6 :
  forall (An:nat -> R) (k:R) (N:nat),
    0 <= k ->
    (forall i:nat, An (S i) < k * An i) ->
    sum_f_R0 An N <= An 0%nat * sum_f_R0 (fun i:nat => k ^ i) N.forall (An : nat -> R) (k : R) (N : nat),
0 <= k ->
(forall i : nat, An (S i) < k * An i) ->
sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
Proof.forall (An : nat -> R) (k : R) (N : nat),
0 <= k ->
(forall i : nat, An (S i) < k * An i) ->
sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
  intros; induction  N as [| N HrecN].An:nat -> R
k:R
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
sum_f_R0 An 0 <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) 0
An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
sum_f_R0 An (S N) <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) (S N)
  simpl; right; ring.An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
sum_f_R0 An (S N) <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) (S N)
  apply Rle_trans with (An 0%nat * sum_f_R0 (fun i:nat => k ^ i) N + An (S N)).An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
sum_f_R0 An (S N) <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N +
An (S N)
An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N +
An (S N) <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) (S N)
  rewrite tech5; do 2 rewrite <- (Rplus_comm (An (S N)));
    apply Rplus_le_compat_l.An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N +
An (S N) <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) (S N)
  apply HrecN.An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N +
An (S N) <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) (S N)
  rewrite tech5; rewrite Rmult_plus_distr_l; apply Rplus_le_compat_l.An:nat -> R
k:R
N:nat
H:0 <= k
H0:forall i : nat, An (S i) < k * An i
HrecN:sum_f_R0 An N <=
An 0%nat * sum_f_R0 (fun i : nat => k ^ i) N
An (S N) <= An 0%nat * k ^ S N
  apply tech4; assumption.
Qed.

Lemma tech7 : forall r1 r2:R, r1 <> 0 -> r2 <> 0 -> r1 <> r2 -> / r1 <> / r2.forall r1 r2 : R,
r1 <> 0 -> r2 <> 0 -> r1 <> r2 -> / r1 <> / r2
Proof.forall r1 r2 : R,
r1 <> 0 -> r2 <> 0 -> r1 <> r2 -> / r1 <> / r2
  intros; red; intro.r1, r2:R
H:r1 <> 0
H0:r2 <> 0
H1:r1 <> r2
H2:/ r1 = / r2
False
  assert (H3 := Rmult_eq_compat_l r1 _ _ H2).r1, r2:R
H:r1 <> 0
H0:r2 <> 0
H1:r1 <> r2
H2:/ r1 = / r2
H3:r1 * / r1 = r1 * / r2
False
  rewrite <- Rinv_r_sym in H3; [ idtac | assumption ].r1, r2:R
H:r1 <> 0
H0:r2 <> 0
H1:r1 <> r2
H2:/ r1 = / r2
H3:1 = r1 * / r2
False
  assert (H4 := Rmult_eq_compat_l r2 _ _ H3).r1, r2:R
H:r1 <> 0
H0:r2 <> 0
H1:r1 <> r2
H2:/ r1 = / r2
H3:1 = r1 * / r2
H4:r2 * 1 = r2 * (r1 * / r2)
False
  rewrite Rmult_1_r in H4; rewrite <- Rmult_assoc in H4.r1, r2:R
H:r1 <> 0
H0:r2 <> 0
H1:r1 <> r2
H2:/ r1 = / r2
H3:1 = r1 * / r2
H4:r2 = r2 * r1 * / r2
False
  rewrite Rinv_r_simpl_m in H4; [ idtac | assumption ].r1, r2:R
H:r1 <> 0
H0:r2 <> 0
H1:r1 <> r2
H2:/ r1 = / r2
H3:1 = r1 * / r2
H4:r2 = r1
False
  elim H1; symmetry ; assumption.
Qed.

Lemma tech11 :
  forall (An Bn Cn:nat -> R) (N:nat),
    (forall i:nat, An i = Bn i - Cn i) ->
    sum_f_R0 An N = sum_f_R0 Bn N - sum_f_R0 Cn N.forall (An Bn Cn : nat -> R) (N : nat),
(forall i : nat, An i = Bn i - Cn i) ->
sum_f_R0 An N = sum_f_R0 Bn N - sum_f_R0 Cn N
Proof.forall (An Bn Cn : nat -> R) (N : nat),
(forall i : nat, An i = Bn i - Cn i) ->
sum_f_R0 An N = sum_f_R0 Bn N - sum_f_R0 Cn N
  intros; induction  N as [| N HrecN].An, Bn, Cn:nat -> R
H:forall i : nat, An i = Bn i - Cn i
sum_f_R0 An 0 = sum_f_R0 Bn 0 - sum_f_R0 Cn 0
An, Bn, Cn:nat -> R
N:nat
H:forall i : nat, An i = Bn i - Cn i
HrecN:sum_f_R0 An N = sum_f_R0 Bn N - sum_f_R0 Cn N
sum_f_R0 An (S N) =
sum_f_R0 Bn (S N) - sum_f_R0 Cn (S N)
  simpl; apply H.An, Bn, Cn:nat -> R
N:nat
H:forall i : nat, An i = Bn i - Cn i
HrecN:sum_f_R0 An N = sum_f_R0 Bn N - sum_f_R0 Cn N
sum_f_R0 An (S N) =
sum_f_R0 Bn (S N) - sum_f_R0 Cn (S N)
  do 3 rewrite tech5; rewrite HrecN; rewrite (H (S N)); ring.
Qed.

Lemma tech12 :
  forall (An:nat -> R) (x l:R),
    Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l ->
    Pser An x l.forall (An : nat -> R) (x l : R),
Un_cv
  (fun N : nat =>
   sum_f_R0 (fun i : nat => An i * x ^ i) N) l ->
Pser An x l
Proof.forall (An : nat -> R) (x l : R),
Un_cv
  (fun N : nat =>
   sum_f_R0 (fun i : nat => An i * x ^ i) N) l ->
Pser An x l
  intros; unfold Pser; unfold infinite_sum; unfold Un_cv in H;
    assumption.
Qed.

Lemma scal_sum :
  forall (An:nat -> R) (N:nat) (x:R),
    x * sum_f_R0 An N = sum_f_R0 (fun i:nat => An i * x) N.forall (An : nat -> R) (N : nat) (x : R),
x * sum_f_R0 An N =
sum_f_R0 (fun i : nat => An i * x) N
Proof.forall (An : nat -> R) (N : nat) (x : R),
x * sum_f_R0 An N =
sum_f_R0 (fun i : nat => An i * x) N
  intros; induction  N as [| N HrecN].An:nat -> R
x:R
x * sum_f_R0 An 0 =
sum_f_R0 (fun i : nat => An i * x) 0
An:nat -> R
N:nat
x:R
HrecN:x * sum_f_R0 An N =
sum_f_R0 (fun i : nat => An i * x) N
x * sum_f_R0 An (S N) =
sum_f_R0 (fun i : nat => An i * x) (S N)
  simpl; ring.An:nat -> R
N:nat
x:R
HrecN:x * sum_f_R0 An N =
sum_f_R0 (fun i : nat => An i * x) N
x * sum_f_R0 An (S N) =
sum_f_R0 (fun i : nat => An i * x) (S N)
  do 2 rewrite tech5.An:nat -> R
N:nat
x:R
HrecN:x * sum_f_R0 An N =
sum_f_R0 (fun i : nat => An i * x) N
x * (sum_f_R0 An N + An (S N)) =
sum_f_R0 (fun i : nat => An i * x) N + An (S N) * x
  rewrite Rmult_plus_distr_l; rewrite <- HrecN; ring.
Qed.

Lemma decomp_sum :
  forall (An:nat -> R) (N:nat),
    (0 < N)%nat ->
    sum_f_R0 An N = An 0%nat + sum_f_R0 (fun i:nat => An (S i)) (pred N).forall (An : nat -> R) (N : nat),
(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i)) (Init.Nat.pred N)
Proof.forall (An : nat -> R) (N : nat),
(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i)) (Init.Nat.pred N)
  intros; induction  N as [| N HrecN].An:nat -> R
H:(0 < 0)%nat
sum_f_R0 An 0 =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i)) (Init.Nat.pred 0)
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
  elim (lt_irrefl _ H).An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
  cut ((0 < N)%nat \/ N = 0%nat).An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat ->
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  intro; elim H0; intro.An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:N = 0%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  cut (S (pred N) = pred (S N)).An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
S (Init.Nat.pred N) = Init.Nat.pred (S N) ->
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
S (Init.Nat.pred N) = Init.Nat.pred (S N)
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:N = 0%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  intro; rewrite <- H2.An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
H2:S (Init.Nat.pred N) = Init.Nat.pred (S N)
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (S (Init.Nat.pred N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
S (Init.Nat.pred N) = Init.Nat.pred (S N)
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:N = 0%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  do 2 rewrite tech5.An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
H2:S (Init.Nat.pred N) = Init.Nat.pred (S N)
sum_f_R0 An N + An (S N) =
An 0%nat +
(sum_f_R0 (fun i : nat => An (S i)) (Init.Nat.pred N) +
 An (S (S (Init.Nat.pred N))))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
S (Init.Nat.pred N) = Init.Nat.pred (S N)
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:N = 0%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  replace (S (S (pred N))) with (S N).An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
H2:S (Init.Nat.pred N) = Init.Nat.pred (S N)
sum_f_R0 An N + An (S N) =
An 0%nat +
(sum_f_R0 (fun i : nat => An (S i)) (Init.Nat.pred N) +
 An (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
H2:S (Init.Nat.pred N) = Init.Nat.pred (S N)
S N = S (S (Init.Nat.pred N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
S (Init.Nat.pred N) = Init.Nat.pred (S N)
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:N = 0%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  rewrite (HrecN H1); ring.An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
H2:S (Init.Nat.pred N) = Init.Nat.pred (S N)
S N = S (S (Init.Nat.pred N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
S (Init.Nat.pred N) = Init.Nat.pred (S N)
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:N = 0%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  rewrite H2; simpl; reflexivity.An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:(0 < N)%nat
S (Init.Nat.pred N) = Init.Nat.pred (S N)
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:N = 0%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  destruct (O_or_S N) as [(m,<-)|<-].An:nat -> R
m:nat
H1:(0 < S m)%nat
H0:(0 < S m)%nat \/ S m = 0%nat
HrecN:(0 < S m)%nat ->
sum_f_R0 An (S m) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S m))
H:(0 < S (S m))%nat
S (Init.Nat.pred (S m)) = Init.Nat.pred (S (S m))
An:nat -> R
H1:(0 < 0)%nat
H0:(0 < 0)%nat \/ 0%nat = 0%nat
HrecN:(0 < 0)%nat ->
sum_f_R0 An 0 =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred 0)
H:(0 < 1)%nat
S (Init.Nat.pred 0) = Init.Nat.pred 1
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:N = 0%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  simpl; reflexivity.An:nat -> R
H1:(0 < 0)%nat
H0:(0 < 0)%nat \/ 0%nat = 0%nat
HrecN:(0 < 0)%nat ->
sum_f_R0 An 0 =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred 0)
H:(0 < 1)%nat
S (Init.Nat.pred 0) = Init.Nat.pred 1
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:N = 0%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  elim (lt_irrefl _ H1).An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H0:(0 < N)%nat \/ N = 0%nat
H1:N = 0%nat
sum_f_R0 An (S N) =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred (S N))
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  rewrite H1; simpl; reflexivity.An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
(0 < N)%nat \/ N = 0%nat
  inversion H.An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
H1:0%nat = N
(0 < 0)%nat \/ 0%nat = 0%nat
An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
m:nat
H1:(1 <= N)%nat
H0:m = N
(0 < N)%nat \/ N = 0%nat
  right; reflexivity.An:nat -> R
N:nat
H:(0 < S N)%nat
HrecN:(0 < N)%nat ->
sum_f_R0 An N =
An 0%nat +
sum_f_R0 (fun i : nat => An (S i))
  (Init.Nat.pred N)
m:nat
H1:(1 <= N)%nat
H0:m = N
(0 < N)%nat \/ N = 0%nat
  left; apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ].
Qed.

Lemma plus_sum :
  forall (An Bn:nat -> R) (N:nat),
    sum_f_R0 (fun i:nat => An i + Bn i) N = sum_f_R0 An N + sum_f_R0 Bn N.forall (An Bn : nat -> R) (N : nat),
sum_f_R0 (fun i : nat => An i + Bn i) N =
sum_f_R0 An N + sum_f_R0 Bn N
Proof.forall (An Bn : nat -> R) (N : nat),
sum_f_R0 (fun i : nat => An i + Bn i) N =
sum_f_R0 An N + sum_f_R0 Bn N
  intros; induction  N as [| N HrecN].An, Bn:nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) 0 =
sum_f_R0 An 0 + sum_f_R0 Bn 0
An, Bn:nat -> R
N:nat
HrecN:sum_f_R0 (fun i : nat => An i + Bn i) N =
sum_f_R0 An N + sum_f_R0 Bn N
sum_f_R0 (fun i : nat => An i + Bn i) (S N) =
sum_f_R0 An (S N) + sum_f_R0 Bn (S N)
  simpl; ring.An, Bn:nat -> R
N:nat
HrecN:sum_f_R0 (fun i : nat => An i + Bn i) N =
sum_f_R0 An N + sum_f_R0 Bn N
sum_f_R0 (fun i : nat => An i + Bn i) (S N) =
sum_f_R0 An (S N) + sum_f_R0 Bn (S N)
  do 3 rewrite tech5; rewrite HrecN; ring.
Qed.

Lemma sum_eq :
  forall (An Bn:nat -> R) (N:nat),
    (forall i:nat, (i <= N)%nat -> An i = Bn i) ->
    sum_f_R0 An N = sum_f_R0 Bn N.forall (An Bn : nat -> R) (N : nat),
(forall i : nat, (i <= N)%nat -> An i = Bn i) ->
sum_f_R0 An N = sum_f_R0 Bn N
Proof.forall (An Bn : nat -> R) (N : nat),
(forall i : nat, (i <= N)%nat -> An i = Bn i) ->
sum_f_R0 An N = sum_f_R0 Bn N
  intros; induction  N as [| N HrecN].An, Bn:nat -> R
H:forall i : nat, (i <= 0)%nat -> An i = Bn i
sum_f_R0 An 0 = sum_f_R0 Bn 0
An, Bn:nat -> R
N:nat
H:forall i : nat, (i <= S N)%nat -> An i = Bn i
HrecN:(forall i : nat, (i <= N)%nat -> An i = Bn i) ->
sum_f_R0 An N = sum_f_R0 Bn N
sum_f_R0 An (S N) = sum_f_R0 Bn (S N)
  simpl; apply H; apply le_n.An, Bn:nat -> R
N:nat
H:forall i : nat, (i <= S N)%nat -> An i = Bn i
HrecN:(forall i : nat, (i <= N)%nat -> An i = Bn i) ->
sum_f_R0 An N = sum_f_R0 Bn N
sum_f_R0 An (S N) = sum_f_R0 Bn (S N)
  do 2 rewrite tech5; rewrite HrecN.An, Bn:nat -> R
N:nat
H:forall i : nat, (i <= S N)%nat -> An i = Bn i
HrecN:(forall i : nat, (i <= N)%nat -> An i = Bn i) ->
sum_f_R0 An N = sum_f_R0 Bn N
sum_f_R0 Bn N + An (S N) = sum_f_R0 Bn N + Bn (S N)
An, Bn:nat -> R
N:nat
H:forall i : nat, (i <= S N)%nat -> An i = Bn i
HrecN:(forall i : nat, (i <= N)%nat -> An i = Bn i) ->
sum_f_R0 An N = sum_f_R0 Bn N
forall i : nat, (i <= N)%nat -> An i = Bn i
  rewrite (H (S N)); [ reflexivity | apply le_n ].An, Bn:nat -> R
N:nat
H:forall i : nat, (i <= S N)%nat -> An i = Bn i
HrecN:(forall i : nat, (i <= N)%nat -> An i = Bn i) ->
sum_f_R0 An N = sum_f_R0 Bn N
forall i : nat, (i <= N)%nat -> An i = Bn i
  intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ].
Qed.

(* Unicity of the limit defined by convergent series *)
Lemma uniqueness_sum :
  forall (An:nat -> R) (l1 l2:R),
    infinite_sum An l1 -> infinite_sum An l2 -> l1 = l2.forall (An : nat -> R) (l1 l2 : R),
infinite_sum An l1 -> infinite_sum An l2 -> l1 = l2
Proof.forall (An : nat -> R) (l1 l2 : R),
infinite_sum An l1 -> infinite_sum An l2 -> l1 = l2
  unfold infinite_sum; intros.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
l1 = l2
  case (Req_dec l1 l2); intro.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 = l2
l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
l1 = l2
  assumption.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
l1 = l2
  cut (0 < Rabs ((l1 - l2) / 2)); [ intro | apply Rabs_pos_lt ].An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  elim (H (Rabs ((l1 - l2) / 2)) H2); intros.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  elim (H0 (Rabs ((l1 - l2) / 2)) H2); intros.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  set (N := max x0 x); cut (N >= x0)%nat.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat -> l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  cut (N >= x)%nat.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat -> (N >= x0)%nat -> l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  intros; assert (H7 := H3 N H5); assert (H8 := H4 N H6).An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  cut (Rabs (l1 - l2) <= R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2).An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2 ->
l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  intro; assert (H10 := Rplus_lt_compat _ _ _ _ H7 H8);
    assert (H11 := Rle_lt_trans _ _ _ H9 H10); unfold Rdiv in H11;
      rewrite Rabs_mult in H11.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
H9:Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 +
R_dist (sum_f_R0 An N) l2
H10:R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2 <
Rabs ((l1 - l2) / 2) + Rabs ((l1 - l2) / 2)
H11:Rabs (l1 - l2) < Rabs (l1 - l2) * Rabs (/ 2) + Rabs (l1 - l2) * Rabs (/ 2)
l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  cut (Rabs (/ 2) = / 2).An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
H9:Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 +
R_dist (sum_f_R0 An N) l2
H10:R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2 <
Rabs ((l1 - l2) / 2) + Rabs ((l1 - l2) / 2)
H11:Rabs (l1 - l2) < Rabs (l1 - l2) * Rabs (/ 2) + Rabs (l1 - l2) * Rabs (/ 2)
Rabs (/ 2) = / 2 -> l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
H9:Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 +
R_dist (sum_f_R0 An N) l2
H10:R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2 <
Rabs ((l1 - l2) / 2) + Rabs ((l1 - l2) / 2)
H11:Rabs (l1 - l2) < Rabs (l1 - l2) * Rabs (/ 2) + Rabs (l1 - l2) * Rabs (/ 2)
Rabs (/ 2) = / 2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  intro; rewrite H12 in H11; assert (H13 := double_var); unfold Rdiv in H13;
    rewrite <- H13 in H11.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
H9:Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 +
R_dist (sum_f_R0 An N) l2
H10:R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2 <
Rabs ((l1 - l2) / 2) + Rabs ((l1 - l2) / 2)
H11:Rabs (l1 - l2) < Rabs (l1 - l2)
H12:Rabs (/ 2) = / 2
H13:forall r1 : R, r1 = r1 * / 2 + r1 * / 2
l1 = l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
H9:Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 +
R_dist (sum_f_R0 An N) l2
H10:R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2 <
Rabs ((l1 - l2) / 2) + Rabs ((l1 - l2) / 2)
H11:Rabs (l1 - l2) < Rabs (l1 - l2) * Rabs (/ 2) + Rabs (l1 - l2) * Rabs (/ 2)
Rabs (/ 2) = / 2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  elim (Rlt_irrefl _ H11).An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
H9:Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 +
R_dist (sum_f_R0 An N) l2
H10:R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2 <
Rabs ((l1 - l2) / 2) + Rabs ((l1 - l2) / 2)
H11:Rabs (l1 - l2) < Rabs (l1 - l2) * Rabs (/ 2) + Rabs (l1 - l2) * Rabs (/ 2)
Rabs (/ 2) = / 2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  apply Rabs_right; left; change (0 < / 2); apply Rinv_0_lt_compat;
    cut (0%nat <> 2%nat);
      [ intro H20; generalize (lt_INR_0 2 (neq_O_lt 2 H20)); unfold INR;
        intro; assumption
        | discriminate ].An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
Rabs (l1 - l2) <=
R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  unfold R_dist; rewrite <- (Rabs_Ropp (sum_f_R0 An N - l1));
    rewrite Ropp_minus_distr'.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
Rabs (l1 - l2) <=
Rabs (l1 - sum_f_R0 An N) + Rabs (sum_f_R0 An N - l2)
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  replace (l1 - l2) with (l1 - sum_f_R0 An N + (sum_f_R0 An N - l2));
  [ idtac | ring ].An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
H5:(N >= x)%nat
H6:(N >= x0)%nat
H7:R_dist (sum_f_R0 An N) l1 < Rabs ((l1 - l2) / 2)
H8:R_dist (sum_f_R0 An N) l2 < Rabs ((l1 - l2) / 2)
Rabs (l1 - sum_f_R0 An N + (sum_f_R0 An N - l2)) <=
Rabs (l1 - sum_f_R0 An N) + Rabs (sum_f_R0 An N - l2)
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  apply Rabs_triang.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  unfold ge; unfold N; apply le_max_r.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
H2:0 < Rabs ((l1 - l2) / 2)
x:nat
H3:forall n : nat,
(n >= x)%nat ->
R_dist (sum_f_R0 An n) l1 < Rabs ((l1 - l2) / 2)
x0:nat
H4:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) l2 < Rabs ((l1 - l2) / 2)
N:=Nat.max x0 x:nat
(N >= x0)%nat
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  unfold ge; unfold N; apply le_max_l.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
(l1 - l2) / 2 <> 0
  unfold Rdiv; apply prod_neq_R0.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
l1 - l2 <> 0
An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
/ 2 <> 0
  apply Rminus_eq_contra; assumption.An:nat -> R
l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:l1 <> l2
/ 2 <> 0
  apply Rinv_neq_0_compat; discrR.
Qed.

Lemma minus_sum :
  forall (An Bn:nat -> R) (N:nat),
    sum_f_R0 (fun i:nat => An i - Bn i) N = sum_f_R0 An N - sum_f_R0 Bn N.forall (An Bn : nat -> R) (N : nat),
sum_f_R0 (fun i : nat => An i - Bn i) N =
sum_f_R0 An N - sum_f_R0 Bn N
Proof.forall (An Bn : nat -> R) (N : nat),
sum_f_R0 (fun i : nat => An i - Bn i) N =
sum_f_R0 An N - sum_f_R0 Bn N
  intros; induction  N as [| N HrecN].An, Bn:nat -> R
sum_f_R0 (fun i : nat => An i - Bn i) 0 =
sum_f_R0 An 0 - sum_f_R0 Bn 0
An, Bn:nat -> R
N:nat
HrecN:sum_f_R0 (fun i : nat => An i - Bn i) N =
sum_f_R0 An N - sum_f_R0 Bn N
sum_f_R0 (fun i : nat => An i - Bn i) (S N) =
sum_f_R0 An (S N) - sum_f_R0 Bn (S N)
  simpl; ring.An, Bn:nat -> R
N:nat
HrecN:sum_f_R0 (fun i : nat => An i - Bn i) N =
sum_f_R0 An N - sum_f_R0 Bn N
sum_f_R0 (fun i : nat => An i - Bn i) (S N) =
sum_f_R0 An (S N) - sum_f_R0 Bn (S N)
  do 3 rewrite tech5; rewrite HrecN; ring.
Qed.

Lemma sum_decomposition :
  forall (An:nat -> R) (N:nat),
    sum_f_R0 (fun l:nat => An (2 * l)%nat) (S N) +
    sum_f_R0 (fun l:nat => An (S (2 * l))) N = sum_f_R0 An (2 * S N).forall (An : nat -> R) (N : nat),
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
Proof.forall (An : nat -> R) (N : nat),
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
  intros.An:nat -> R
N:nat
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
  induction  N as [| N HrecN].An:nat -> R
sum_f_R0 (fun l : nat => An (2 * l)%nat) 1 +
sum_f_R0 (fun l : nat => An (S (2 * l))) 0 =
sum_f_R0 An (2 * 1)
An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S (S N)) +
sum_f_R0 (fun l : nat => An (S (2 * l))) (S N) =
sum_f_R0 An (2 * S (S N))
  simpl; ring.An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S (S N)) +
sum_f_R0 (fun l : nat => An (S (2 * l))) (S N) =
sum_f_R0 An (2 * S (S N))
  rewrite tech5.An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
An (2 * S (S N))%nat +
sum_f_R0 (fun l : nat => An (S (2 * l))) (S N) =
sum_f_R0 An (2 * S (S N))
  rewrite (tech5 (fun l:nat => An (S (2 * l))) N).An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
An (2 * S (S N))%nat +
(sum_f_R0 (fun l : nat => An (S (2 * l))) N +
 An (S (2 * S N))) = sum_f_R0 An (2 * S (S N))
  replace (2 * S (S N))%nat with (S (S (2 * S N))).An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
An (S (S (2 * S N))) +
(sum_f_R0 (fun l : nat => An (S (2 * l))) N +
 An (S (2 * S N))) = sum_f_R0 An (S (S (2 * S N)))
An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
S (S (2 * S N)) = (2 * S (S N))%nat
  rewrite (tech5 An (S (2 * S N))).An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
An (S (S (2 * S N))) +
(sum_f_R0 (fun l : nat => An (S (2 * l))) N +
 An (S (2 * S N))) =
sum_f_R0 An (S (2 * S N)) + An (S (S (2 * S N)))
An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
S (S (2 * S N)) = (2 * S (S N))%nat
  rewrite (tech5 An (2 * S N)).An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
An (S (S (2 * S N))) +
(sum_f_R0 (fun l : nat => An (S (2 * l))) N +
 An (S (2 * S N))) =
sum_f_R0 An (2 * S N) + An (S (2 * S N)) +
An (S (S (2 * S N)))
An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
S (S (2 * S N)) = (2 * S (S N))%nat
  rewrite <- HrecN.An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
An (S (S (2 * S N))) +
(sum_f_R0 (fun l : nat => An (S (2 * l))) N +
 An (S (2 * S N))) =
sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N +
An (S (2 * S N)) + An (S (S (2 * S N)))
An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
S (S (2 * S N)) = (2 * S (S N))%nat
  ring.An:nat -> R
N:nat
HrecN:sum_f_R0 (fun l : nat => An (2 * l)%nat) (S N) +
sum_f_R0 (fun l : nat => An (S (2 * l))) N =
sum_f_R0 An (2 * S N)
S (S (2 * S N)) = (2 * S (S N))%nat
  ring.
Qed.

Lemma sum_Rle :
  forall (An Bn:nat -> R) (N:nat),
    (forall n:nat, (n <= N)%nat -> An n <= Bn n) ->
    sum_f_R0 An N <= sum_f_R0 Bn N.forall (An Bn : nat -> R) (N : nat),
(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
Proof.forall (An Bn : nat -> R) (N : nat),
(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
  intros.An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= N)%nat -> An n <= Bn n
sum_f_R0 An N <= sum_f_R0 Bn N
  induction  N as [| N HrecN].An, Bn:nat -> R
H:forall n : nat, (n <= 0)%nat -> An n <= Bn n
sum_f_R0 An 0 <= sum_f_R0 Bn 0
An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An (S N) <= sum_f_R0 Bn (S N)
  simpl; apply H.An, Bn:nat -> R
H:forall n : nat, (n <= 0)%nat -> An n <= Bn n
(0 <= 0)%nat
An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An (S N) <= sum_f_R0 Bn (S N)
  apply le_n.An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An (S N) <= sum_f_R0 Bn (S N)
  do 2 rewrite tech5.An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N + An (S N) <= sum_f_R0 Bn N + Bn (S N)
  apply Rle_trans with (sum_f_R0 An N + Bn (S N)).An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N + An (S N) <= sum_f_R0 An N + Bn (S N)
An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N + Bn (S N) <= sum_f_R0 Bn N + Bn (S N)
  apply Rplus_le_compat_l.An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
An (S N) <= Bn (S N)
An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N + Bn (S N) <= sum_f_R0 Bn N + Bn (S N)
  apply H.An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
(S N <= S N)%nat
An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N + Bn (S N) <= sum_f_R0 Bn N + Bn (S N)
  apply le_n.An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N + Bn (S N) <= sum_f_R0 Bn N + Bn (S N)
  do 2 rewrite <- (Rplus_comm (Bn (S N))).An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
Bn (S N) + sum_f_R0 An N <= Bn (S N) + sum_f_R0 Bn N
  apply Rplus_le_compat_l.An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N <= sum_f_R0 Bn N
  apply HrecN.An, Bn:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n <= Bn n
HrecN:(forall n : nat, (n <= N)%nat -> An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
forall n : nat, (n <= N)%nat -> An n <= Bn n
  intros; apply H.An, Bn:nat -> R
N:nat
H:forall n0 : nat, (n0 <= S N)%nat -> An n0 <= Bn n0
HrecN:(forall n0 : nat,
 (n0 <= N)%nat -> An n0 <= Bn n0) ->
sum_f_R0 An N <= sum_f_R0 Bn N
n:nat
H0:(n <= N)%nat
(n <= S N)%nat
  apply le_trans with N; [ assumption | apply le_n_Sn ].
Qed.

Lemma Rsum_abs :
  forall (An:nat -> R) (N:nat),
    Rabs (sum_f_R0 An N) <= sum_f_R0 (fun l:nat => Rabs (An l)) N.forall (An : nat -> R) (N : nat),
Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
Proof.forall (An : nat -> R) (N : nat),
Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
  intros.An:nat -> R
N:nat
Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
  induction  N as [| N HrecN].An:nat -> R
Rabs (sum_f_R0 An 0) <=
sum_f_R0 (fun l : nat => Rabs (An l)) 0
An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
Rabs (sum_f_R0 An (S N)) <=
sum_f_R0 (fun l : nat => Rabs (An l)) (S N)
  simpl.An:nat -> R
Rabs (An 0%nat) <= Rabs (An 0%nat)
An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
Rabs (sum_f_R0 An (S N)) <=
sum_f_R0 (fun l : nat => Rabs (An l)) (S N)
  right; reflexivity.An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
Rabs (sum_f_R0 An (S N)) <=
sum_f_R0 (fun l : nat => Rabs (An l)) (S N)
  do 2 rewrite tech5.An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
Rabs (sum_f_R0 An N + An (S N)) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N +
Rabs (An (S N))
  apply Rle_trans with (Rabs (sum_f_R0 An N) + Rabs (An (S N))).An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
Rabs (sum_f_R0 An N + An (S N)) <=
Rabs (sum_f_R0 An N) + Rabs (An (S N))
An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
Rabs (sum_f_R0 An N) + Rabs (An (S N)) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N +
Rabs (An (S N))
  apply Rabs_triang.An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
Rabs (sum_f_R0 An N) + Rabs (An (S N)) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N +
Rabs (An (S N))
  do 2 rewrite <- (Rplus_comm (Rabs (An (S N)))).An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
Rabs (An (S N)) + Rabs (sum_f_R0 An N) <=
Rabs (An (S N)) +
sum_f_R0 (fun l : nat => Rabs (An l)) N
  apply Rplus_le_compat_l.An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun l : nat => Rabs (An l)) N
  apply HrecN.
Qed.

Lemma sum_cte :
  forall (x:R) (N:nat), sum_f_R0 (fun _:nat => x) N = x * INR (S N).forall (x : R) (N : nat),
sum_f_R0 (fun _ : nat => x) N = x * INR (S N)
Proof.forall (x : R) (N : nat),
sum_f_R0 (fun _ : nat => x) N = x * INR (S N)
  intros.x:R
N:nat
sum_f_R0 (fun _ : nat => x) N = x * INR (S N)
  induction  N as [| N HrecN].x:R
sum_f_R0 (fun _ : nat => x) 0 = x * INR 1
x:R
N:nat
HrecN:sum_f_R0 (fun _ : nat => x) N = x * INR (S N)
sum_f_R0 (fun _ : nat => x) (S N) = x * INR (S (S N))
  simpl; ring.x:R
N:nat
HrecN:sum_f_R0 (fun _ : nat => x) N = x * INR (S N)
sum_f_R0 (fun _ : nat => x) (S N) = x * INR (S (S N))
  rewrite tech5.x:R
N:nat
HrecN:sum_f_R0 (fun _ : nat => x) N = x * INR (S N)
sum_f_R0 (fun _ : nat => x) N + x = x * INR (S (S N))
  rewrite HrecN; repeat rewrite S_INR; ring.
Qed.

(**********)
Lemma sum_growing :
  forall (An Bn:nat -> R) (N:nat),
    (forall n:nat, An n <= Bn n) -> sum_f_R0 An N <= sum_f_R0 Bn N.forall (An Bn : nat -> R) (N : nat),
(forall n : nat, An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
Proof.forall (An Bn : nat -> R) (N : nat),
(forall n : nat, An n <= Bn n) ->
sum_f_R0 An N <= sum_f_R0 Bn N
  intros.An, Bn:nat -> R
N:nat
H:forall n : nat, An n <= Bn n
sum_f_R0 An N <= sum_f_R0 Bn N
  induction  N as [| N HrecN].An, Bn:nat -> R
H:forall n : nat, An n <= Bn n
sum_f_R0 An 0 <= sum_f_R0 Bn 0
An, Bn:nat -> R
N:nat
H:forall n : nat, An n <= Bn n
HrecN:sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An (S N) <= sum_f_R0 Bn (S N)
  simpl; apply H.An, Bn:nat -> R
N:nat
H:forall n : nat, An n <= Bn n
HrecN:sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An (S N) <= sum_f_R0 Bn (S N)
  do 2 rewrite tech5.An, Bn:nat -> R
N:nat
H:forall n : nat, An n <= Bn n
HrecN:sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N + An (S N) <= sum_f_R0 Bn N + Bn (S N)
  apply Rle_trans with (sum_f_R0 An N + Bn (S N)).An, Bn:nat -> R
N:nat
H:forall n : nat, An n <= Bn n
HrecN:sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N + An (S N) <= sum_f_R0 An N + Bn (S N)
An, Bn:nat -> R
N:nat
H:forall n : nat, An n <= Bn n
HrecN:sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N + Bn (S N) <= sum_f_R0 Bn N + Bn (S N)
  apply Rplus_le_compat_l; apply H.An, Bn:nat -> R
N:nat
H:forall n : nat, An n <= Bn n
HrecN:sum_f_R0 An N <= sum_f_R0 Bn N
sum_f_R0 An N + Bn (S N) <= sum_f_R0 Bn N + Bn (S N)
  do 2 rewrite <- (Rplus_comm (Bn (S N))).An, Bn:nat -> R
N:nat
H:forall n : nat, An n <= Bn n
HrecN:sum_f_R0 An N <= sum_f_R0 Bn N
Bn (S N) + sum_f_R0 An N <= Bn (S N) + sum_f_R0 Bn N
  apply Rplus_le_compat_l; apply HrecN.
Qed.

(**********)
Lemma Rabs_triang_gen :
  forall (An:nat -> R) (N:nat),
    Rabs (sum_f_R0 An N) <= sum_f_R0 (fun i:nat => Rabs (An i)) N.forall (An : nat -> R) (N : nat),
Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
Proof.forall (An : nat -> R) (N : nat),
Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
  intros.An:nat -> R
N:nat
Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
  induction  N as [| N HrecN].An:nat -> R
Rabs (sum_f_R0 An 0) <=
sum_f_R0 (fun i : nat => Rabs (An i)) 0
An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
Rabs (sum_f_R0 An (S N)) <=
sum_f_R0 (fun i : nat => Rabs (An i)) (S N)
  simpl.An:nat -> R
Rabs (An 0%nat) <= Rabs (An 0%nat)
An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
Rabs (sum_f_R0 An (S N)) <=
sum_f_R0 (fun i : nat => Rabs (An i)) (S N)
  right; reflexivity.An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
Rabs (sum_f_R0 An (S N)) <=
sum_f_R0 (fun i : nat => Rabs (An i)) (S N)
  do 2 rewrite tech5.An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
Rabs (sum_f_R0 An N + An (S N)) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N +
Rabs (An (S N))
  apply Rle_trans with (Rabs (sum_f_R0 An N) + Rabs (An (S N))).An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
Rabs (sum_f_R0 An N + An (S N)) <=
Rabs (sum_f_R0 An N) + Rabs (An (S N))
An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
Rabs (sum_f_R0 An N) + Rabs (An (S N)) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N +
Rabs (An (S N))
  apply Rabs_triang.An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
Rabs (sum_f_R0 An N) + Rabs (An (S N)) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N +
Rabs (An (S N))
  do 2 rewrite <- (Rplus_comm (Rabs (An (S N)))).An:nat -> R
N:nat
HrecN:Rabs (sum_f_R0 An N) <=
sum_f_R0 (fun i : nat => Rabs (An i)) N
Rabs (An (S N)) + Rabs (sum_f_R0 An N) <=
Rabs (An (S N)) +
sum_f_R0 (fun i : nat => Rabs (An i)) N
  apply Rplus_le_compat_l; apply HrecN.
Qed.

(**********)
Lemma cond_pos_sum :
  forall (An:nat -> R) (N:nat),
    (forall n:nat, 0 <= An n) -> 0 <= sum_f_R0 An N.forall (An : nat -> R) (N : nat),
(forall n : nat, 0 <= An n) -> 0 <= sum_f_R0 An N
Proof.forall (An : nat -> R) (N : nat),
(forall n : nat, 0 <= An n) -> 0 <= sum_f_R0 An N
  intros.An:nat -> R
N:nat
H:forall n : nat, 0 <= An n
0 <= sum_f_R0 An N
  induction  N as [| N HrecN].An:nat -> R
H:forall n : nat, 0 <= An n
0 <= sum_f_R0 An 0
An:nat -> R
N:nat
H:forall n : nat, 0 <= An n
HrecN:0 <= sum_f_R0 An N
0 <= sum_f_R0 An (S N)
  simpl; apply H.An:nat -> R
N:nat
H:forall n : nat, 0 <= An n
HrecN:0 <= sum_f_R0 An N
0 <= sum_f_R0 An (S N)
  rewrite tech5.An:nat -> R
N:nat
H:forall n : nat, 0 <= An n
HrecN:0 <= sum_f_R0 An N
0 <= sum_f_R0 An N + An (S N)
  apply Rplus_le_le_0_compat.An:nat -> R
N:nat
H:forall n : nat, 0 <= An n
HrecN:0 <= sum_f_R0 An N
0 <= sum_f_R0 An N
An:nat -> R
N:nat
H:forall n : nat, 0 <= An n
HrecN:0 <= sum_f_R0 An N
0 <= An (S N)
  apply HrecN.An:nat -> R
N:nat
H:forall n : nat, 0 <= An n
HrecN:0 <= sum_f_R0 An N
0 <= An (S N)
  apply H.
Qed.

(* Cauchy's criterion for series *)
Definition Cauchy_crit_series (An:nat -> R) : Prop :=
  Cauchy_crit (fun N:nat => sum_f_R0 An N).

(* If (|An|) satisfies the Cauchy's criterion for series, then (An) too *)
Lemma cauchy_abs :
  forall An:nat -> R,
    Cauchy_crit_series (fun i:nat => Rabs (An i)) -> Cauchy_crit_series An.forall An : nat -> R,
Cauchy_crit_series (fun i : nat => Rabs (An i)) ->
Cauchy_crit_series An
Proof.forall An : nat -> R,
Cauchy_crit_series (fun i : nat => Rabs (An i)) ->
Cauchy_crit_series An
  unfold Cauchy_crit_series; unfold Cauchy_crit.forall An : nat -> R,
(forall eps : R,
 eps > 0 ->
 exists N : nat,
   forall n m : nat,
   (n >= N)%nat ->
   (m >= N)%nat ->
   R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
     (sum_f_R0 (fun i : nat => Rabs (An i)) m) < eps) ->
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
  intros.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m) <
  eps0
eps:R
H0:eps > 0
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
  elim (H eps H0); intros.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m : nat,
(n >= x)%nat ->
(m >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m) < eps
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
  exists x.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m : nat,
(n >= x)%nat ->
(m >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m) < eps
forall n m : nat,
(n >= x)%nat ->
(m >= x)%nat ->
R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
  intros.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
  cut
    (R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
      R_dist (sum_f_R0 (fun i:nat => Rabs (An i)) n)
      (sum_f_R0 (fun i:nat => Rabs (An i)) m)).An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m) ->
R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  intro.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
H4:R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  apply Rle_lt_trans with
    (R_dist (sum_f_R0 (fun i:nat => Rabs (An i)) n)
      (sum_f_R0 (fun i:nat => Rabs (An i)) m)).An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
H4:R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
H4:R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m) < eps
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  assumption.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
H4:R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m) < eps
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  apply H1; assumption.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  destruct (lt_eq_lt_dec n m) as [[ | -> ]|].An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  rewrite (tech2 An n m); [ idtac | assumption ].An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
R_dist (sum_f_R0 An n)
  (sum_f_R0 An n +
   sum_f_R0 (fun i : nat => An (S n + i)%nat)
     (m - S n)) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  rewrite (tech2 (fun i:nat => Rabs (An i)) n m); [ idtac | assumption ].An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
R_dist (sum_f_R0 An n)
  (sum_f_R0 An n +
   sum_f_R0 (fun i : nat => An (S n + i)%nat)
     (m - S n)) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) n +
   sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
     (m - S n))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  unfold R_dist.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Rabs
  (sum_f_R0 An n -
   (sum_f_R0 An n +
    sum_f_R0 (fun i : nat => An (S n + i)%nat)
      (m - S n))) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An i)) n -
   (sum_f_R0 (fun i : nat => Rabs (An i)) n +
    sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
      (m - S n)))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  unfold Rminus.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Rabs
  (sum_f_R0 An n +
   -
   (sum_f_R0 An n +
    sum_f_R0 (fun i : nat => An (S n + i)%nat)
      (m - S n))) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An i)) n +
   -
   (sum_f_R0 (fun i : nat => Rabs (An i)) n +
    sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
      (m - S n)))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  do 2 rewrite Ropp_plus_distr.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Rabs
  (sum_f_R0 An n +
   (- sum_f_R0 An n +
    -
    sum_f_R0 (fun i : nat => An (S n + i)%nat)
      (m - S n))) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An i)) n +
   (- sum_f_R0 (fun i : nat => Rabs (An i)) n +
    -
    sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
      (m - S n)))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  do 2 rewrite <- Rplus_assoc.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Rabs
  (sum_f_R0 An n + - sum_f_R0 An n +
   -
   sum_f_R0 (fun i : nat => An (S n + i)%nat)
     (m - S n)) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An i)) n +
   - sum_f_R0 (fun i : nat => Rabs (An i)) n +
   -
   sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
     (m - S n))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  do 2 rewrite Rplus_opp_r.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Rabs
  (0 +
   -
   sum_f_R0 (fun i : nat => An (S n + i)%nat)
     (m - S n)) <=
Rabs
  (0 +
   -
   sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
     (m - S n))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  do 2 rewrite Rplus_0_l.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Rabs
  (-
   sum_f_R0 (fun i : nat => An (S n + i)%nat)
     (m - S n)) <=
Rabs
  (-
   sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
     (m - S n))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  do 2 rewrite Rabs_Ropp.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Rabs
  (sum_f_R0 (fun i : nat => An (S n + i)%nat)
     (m - S n)) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
     (m - S n))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  rewrite
    (Rabs_right (sum_f_R0 (fun i:nat => Rabs (An (S n + i)%nat)) (m - S n)))
    .An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Rabs
  (sum_f_R0 (fun i : nat => An (S n + i)%nat)
     (m - S n)) <=
sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
  (m - S n)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
  (m - S n) >= 0
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  set (Bn := fun i:nat => An (S n + i)%nat).An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Bn:=fun i : nat => An (S n + i)%nat:nat -> R
Rabs (sum_f_R0 Bn (m - S n)) <=
sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
  (m - S n)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
  (m - S n) >= 0
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  replace (fun i:nat => Rabs (An (S n + i)%nat)) with
  (fun i:nat => Rabs (Bn i)).An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Bn:=fun i : nat => An (S n + i)%nat:nat -> R
Rabs (sum_f_R0 Bn (m - S n)) <=
sum_f_R0 (fun i : nat => Rabs (Bn i)) (m - S n)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Bn:=fun i : nat => An (S n + i)%nat:nat -> R
(fun i : nat => Rabs (Bn i)) =
(fun i : nat => Rabs (An (S n + i)%nat))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
  (m - S n) >= 0
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  apply Rabs_triang_gen.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
Bn:=fun i : nat => An (S n + i)%nat:nat -> R
(fun i : nat => Rabs (Bn i)) =
(fun i : nat => Rabs (An (S n + i)%nat))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
  (m - S n) >= 0
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  unfold Bn; reflexivity.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
  (m - S n) >= 0
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  apply Rle_ge.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
0 <=
sum_f_R0 (fun i : nat => Rabs (An (S n + i)%nat))
  (m - S n)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  apply cond_pos_sum.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(n < m)%nat
forall n0 : nat, 0 <= Rabs (An (S n + n0)%nat)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  intro; apply Rabs_pos.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  unfold R_dist.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
Rabs (sum_f_R0 An m - sum_f_R0 An m) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An i)) m -
   sum_f_R0 (fun i : nat => Rabs (An i)) m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  unfold Rminus; do 2 rewrite Rplus_opp_r.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n m0 : nat,
  (n >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
m:nat
H2, H3:(m >= x)%nat
Rabs 0 <= Rabs 0
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  rewrite Rabs_R0; right; reflexivity.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  rewrite (tech2 An m n); [ idtac | assumption ].An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist
  (sum_f_R0 An m +
   sum_f_R0 (fun i : nat => An (S m + i)%nat)
     (n - S m)) (sum_f_R0 An m) <=
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  rewrite (tech2 (fun i:nat => Rabs (An i)) m n); [ idtac | assumption ].An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
R_dist
  (sum_f_R0 An m +
   sum_f_R0 (fun i : nat => An (S m + i)%nat)
     (n - S m)) (sum_f_R0 An m) <=
R_dist
  (sum_f_R0 (fun i : nat => Rabs (An i)) m +
   sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
     (n - S m))
  (sum_f_R0 (fun i : nat => Rabs (An i)) m)
  unfold R_dist.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Rabs
  (sum_f_R0 An m +
   sum_f_R0 (fun i : nat => An (S m + i)%nat)
     (n - S m) - sum_f_R0 An m) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An i)) m +
   sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
     (n - S m) -
   sum_f_R0 (fun i : nat => Rabs (An i)) m)
  unfold Rminus.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Rabs
  (sum_f_R0 An m +
   sum_f_R0 (fun i : nat => An (S m + i)%nat)
     (n - S m) + - sum_f_R0 An m) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An i)) m +
   sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
     (n - S m) +
   - sum_f_R0 (fun i : nat => Rabs (An i)) m)
  do 2 rewrite Rplus_assoc.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Rabs
  (sum_f_R0 An m +
   (sum_f_R0 (fun i : nat => An (S m + i)%nat)
      (n - S m) + - sum_f_R0 An m)) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An i)) m +
   (sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
      (n - S m) +
    - sum_f_R0 (fun i : nat => Rabs (An i)) m))
  rewrite (Rplus_comm (sum_f_R0 An m)).An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Rabs
  (sum_f_R0 (fun i : nat => An (S m + i)%nat)
     (n - S m) + - sum_f_R0 An m + sum_f_R0 An m) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An i)) m +
   (sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
      (n - S m) +
    - sum_f_R0 (fun i : nat => Rabs (An i)) m))
  rewrite (Rplus_comm (sum_f_R0 (fun i:nat => Rabs (An i)) m)).An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Rabs
  (sum_f_R0 (fun i : nat => An (S m + i)%nat)
     (n - S m) + - sum_f_R0 An m + sum_f_R0 An m) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
     (n - S m) +
   - sum_f_R0 (fun i : nat => Rabs (An i)) m +
   sum_f_R0 (fun i : nat => Rabs (An i)) m)
  do 2 rewrite Rplus_assoc.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Rabs
  (sum_f_R0 (fun i : nat => An (S m + i)%nat)
     (n - S m) + (- sum_f_R0 An m + sum_f_R0 An m)) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
     (n - S m) +
   (- sum_f_R0 (fun i : nat => Rabs (An i)) m +
    sum_f_R0 (fun i : nat => Rabs (An i)) m))
  do 2 rewrite Rplus_opp_l.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Rabs
  (sum_f_R0 (fun i : nat => An (S m + i)%nat)
     (n - S m) + 0) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
     (n - S m) + 0)
  do 2 rewrite Rplus_0_r.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Rabs
  (sum_f_R0 (fun i : nat => An (S m + i)%nat)
     (n - S m)) <=
Rabs
  (sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
     (n - S m))
  rewrite
    (Rabs_right (sum_f_R0 (fun i:nat => Rabs (An (S m + i)%nat)) (n - S m)))
    .An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Rabs
  (sum_f_R0 (fun i : nat => An (S m + i)%nat)
     (n - S m)) <=
sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
  (n - S m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
  (n - S m) >= 0
  set (Bn := fun i:nat => An (S m + i)%nat).An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Bn:=fun i : nat => An (S m + i)%nat:nat -> R
Rabs (sum_f_R0 Bn (n - S m)) <=
sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
  (n - S m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
  (n - S m) >= 0
  replace (fun i:nat => Rabs (An (S m + i)%nat)) with
  (fun i:nat => Rabs (Bn i)).An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Bn:=fun i : nat => An (S m + i)%nat:nat -> R
Rabs (sum_f_R0 Bn (n - S m)) <=
sum_f_R0 (fun i : nat => Rabs (Bn i)) (n - S m)
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Bn:=fun i : nat => An (S m + i)%nat:nat -> R
(fun i : nat => Rabs (Bn i)) =
(fun i : nat => Rabs (An (S m + i)%nat))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
  (n - S m) >= 0
  apply Rabs_triang_gen.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
Bn:=fun i : nat => An (S m + i)%nat:nat -> R
(fun i : nat => Rabs (Bn i)) =
(fun i : nat => Rabs (An (S m + i)%nat))
An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
  (n - S m) >= 0
  unfold Bn; reflexivity.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
  (n - S m) >= 0
  apply Rle_ge.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
0 <=
sum_f_R0 (fun i : nat => Rabs (An (S m + i)%nat))
  (n - S m)
  apply cond_pos_sum.An:nat -> R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 m0 : nat,
  (n0 >= N)%nat ->
  (m0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
    (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
  eps0
eps:R
H0:eps > 0
x:nat
H1:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 (fun i : nat => Rabs (An i)) n0)
  (sum_f_R0 (fun i : nat => Rabs (An i)) m0) <
eps
n, m:nat
H2:(n >= x)%nat
H3:(m >= x)%nat
l:(m < n)%nat
forall n0 : nat, 0 <= Rabs (An (S m + n0)%nat)
  intro; apply Rabs_pos.
Qed.

(**********)
Lemma cv_cauchy_1 :
  forall An:nat -> R,
    { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l } ->
    Cauchy_crit_series An.forall An : nat -> R,
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} ->
Cauchy_crit_series An
Proof.forall An : nat -> R,
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l} ->
Cauchy_crit_series An
  intros An (x,p).An:nat -> R
x:R
p:Un_cv (fun N : nat => sum_f_R0 An N) x
Cauchy_crit_series An
  unfold Un_cv in p.An:nat -> R
x:R
p:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps
Cauchy_crit_series An
  unfold Cauchy_crit_series; unfold Cauchy_crit.An:nat -> R
x:R
p:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
  intros.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
  cut (0 < eps / 2).An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2 ->
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  intro.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  elim (p (eps / 2) H0); intros.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) x < eps / 2
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  exists x0.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n : nat,
(n >= x0)%nat ->
R_dist (sum_f_R0 An n) x < eps / 2
forall n m : nat,
(n >= x0)%nat ->
(m >= x0)%nat ->
R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  intros.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  apply Rle_lt_trans with (R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x).An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x <
eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  unfold R_dist.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
Rabs (sum_f_R0 An n - sum_f_R0 An m) <=
Rabs (sum_f_R0 An n - x) + Rabs (sum_f_R0 An m - x)
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x <
eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  replace (sum_f_R0 An n - sum_f_R0 An m) with
  (sum_f_R0 An n - x + - (sum_f_R0 An m - x)); [ idtac | ring ].An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
Rabs (sum_f_R0 An n - x + - (sum_f_R0 An m - x)) <=
Rabs (sum_f_R0 An n - x) + Rabs (sum_f_R0 An m - x)
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x <
eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  rewrite <- (Rabs_Ropp (sum_f_R0 An m - x)).An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
Rabs (sum_f_R0 An n - x + - (sum_f_R0 An m - x)) <=
Rabs (sum_f_R0 An n - x) +
Rabs (- (sum_f_R0 An m - x))
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x <
eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  apply Rabs_triang.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x <
eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  apply Rlt_le_trans with (eps / 2 + eps / 2).An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x <
eps / 2 + eps / 2
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
eps / 2 + eps / 2 <= eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  apply Rplus_lt_compat.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An n) x < eps / 2
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An m) x < eps / 2
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
eps / 2 + eps / 2 <= eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  apply H1; assumption.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
R_dist (sum_f_R0 An m) x < eps / 2
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
eps / 2 + eps / 2 <= eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  apply H1; assumption.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (sum_f_R0 An n0) x < eps0
eps:R
H:eps > 0
H0:0 < eps / 2
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (sum_f_R0 An n0) x < eps / 2
n, m:nat
H2:(n >= x0)%nat
H3:(m >= x0)%nat
eps / 2 + eps / 2 <= eps
An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  right; symmetry ; apply double_var.An:nat -> R
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) x < eps0
eps:R
H:eps > 0
0 < eps / 2
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
Qed.

Lemma cv_cauchy_2 :
  forall An:nat -> R,
    Cauchy_crit_series An ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.forall An : nat -> R,
Cauchy_crit_series An ->
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Proof.forall An : nat -> R,
Cauchy_crit_series An ->
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
  intros.An:nat -> R
H:Cauchy_crit_series An
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
  apply R_complete.An:nat -> R
H:Cauchy_crit_series An
Cauchy_crit (fun N : nat => sum_f_R0 An N)
  unfold Cauchy_crit_series in H.An:nat -> R
H:Cauchy_crit (fun N : nat => sum_f_R0 An N)
Cauchy_crit (fun N : nat => sum_f_R0 An N)
  exact H.
Qed.

(**********)
Lemma sum_eq_R0 :
  forall (An:nat -> R) (N:nat),
    (forall n:nat, (n <= N)%nat -> An n = 0) -> sum_f_R0 An N = 0.forall (An : nat -> R) (N : nat),
(forall n : nat, (n <= N)%nat -> An n = 0) ->
sum_f_R0 An N = 0
Proof.forall (An : nat -> R) (N : nat),
(forall n : nat, (n <= N)%nat -> An n = 0) ->
sum_f_R0 An N = 0
  intros; induction  N as [| N HrecN].An:nat -> R
H:forall n : nat, (n <= 0)%nat -> An n = 0
sum_f_R0 An 0 = 0
An:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n = 0
HrecN:(forall n : nat, (n <= N)%nat -> An n = 0) ->
sum_f_R0 An N = 0
sum_f_R0 An (S N) = 0
  simpl; apply H; apply le_n.An:nat -> R
N:nat
H:forall n : nat, (n <= S N)%nat -> An n = 0
HrecN:(forall n : nat, (n <= N)%nat -> An n = 0) ->
sum_f_R0 An N = 0
sum_f_R0 An (S N) = 0
  rewrite tech5; rewrite HrecN;
    [ rewrite Rplus_0_l; apply H; apply le_n
      | intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ] ].
Qed.

Definition SP (fn:nat -> R -> R) (N:nat) (x:R) : R :=
  sum_f_R0 (fun k:nat => fn k x) N.

(**********)
Lemma sum_incr :
  forall (An:nat -> R) (N:nat) (l:R),
    Un_cv (fun n:nat => sum_f_R0 An n) l ->
    (forall n:nat, 0 <= An n) -> sum_f_R0 An N <= l.forall (An : nat -> R) (N : nat) (l : R),
Un_cv (fun n : nat => sum_f_R0 An n) l ->
(forall n : nat, 0 <= An n) -> sum_f_R0 An N <= l
Proof.forall (An : nat -> R) (N : nat) (l : R),
Un_cv (fun n : nat => sum_f_R0 An n) l ->
(forall n : nat, 0 <= An n) -> sum_f_R0 An N <= l
  intros; destruct (total_order_T (sum_f_R0 An N) l) as [[Hlt|Heq]|Hgt].An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hlt:sum_f_R0 An N < l
sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Heq:sum_f_R0 An N = l
sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
sum_f_R0 An N <= l
  left; apply Hlt.An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Heq:sum_f_R0 An N = l
sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
sum_f_R0 An N <= l
  right; apply Heq.An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
sum_f_R0 An N <= l
  cut (Un_growing (fun n:nat => sum_f_R0 An n)).An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n) ->
sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  intro; set (l1 := sum_f_R0 An N) in Hgt.An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  unfold Un_cv in H; cut (0 < l1 - l).An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l -> sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  intro; elim (H _ H2); intros.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  set (N0 := max x N); cut (N0 >= x)%nat.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat -> sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  intro; assert (H5 := H3 N0 H4).An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  cut (l1 <= sum_f_R0 An N0).An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0 -> sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  intro; unfold R_dist in H5; rewrite Rabs_right in H5.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:sum_f_R0 An N0 - l < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:Rabs (sum_f_R0 An N0 - l) < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N0 - l >= 0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  cut (sum_f_R0 An N0 < l1).An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:sum_f_R0 An N0 - l < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N0 < l1 -> sum_f_R0 An N <= l
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:sum_f_R0 An N0 - l < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N0 < l1
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:Rabs (sum_f_R0 An N0 - l) < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N0 - l >= 0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  intro; elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H7 H6)).An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:sum_f_R0 An N0 - l < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N0 < l1
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:Rabs (sum_f_R0 An N0 - l) < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N0 - l >= 0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  apply Rplus_lt_reg_l with (- l).An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:sum_f_R0 An N0 - l < l1 - l
H6:l1 <= sum_f_R0 An N0
- l + sum_f_R0 An N0 < - l + l1
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:Rabs (sum_f_R0 An N0 - l) < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N0 - l >= 0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  do 2 rewrite (Rplus_comm (- l)).An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:sum_f_R0 An N0 - l < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N0 + - l < l1 + - l
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:Rabs (sum_f_R0 An N0 - l) < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N0 - l >= 0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  apply H5.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:Rabs (sum_f_R0 An N0 - l) < l1 - l
H6:l1 <= sum_f_R0 An N0
sum_f_R0 An N0 - l >= 0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  apply Rle_ge; apply Rplus_le_reg_l with l.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:Rabs (sum_f_R0 An N0 - l) < l1 - l
H6:l1 <= sum_f_R0 An N0
l + 0 <= l + (sum_f_R0 An N0 - l)
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  rewrite Rplus_0_r; replace (l + (sum_f_R0 An N0 - l)) with (sum_f_R0 An N0);
    [ idtac | ring ]; apply Rle_trans with l1.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:Rabs (sum_f_R0 An N0 - l) < l1 - l
H6:l1 <= sum_f_R0 An N0
l <= l1
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:Rabs (sum_f_R0 An N0 - l) < l1 - l
H6:l1 <= sum_f_R0 An N0
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  left; apply Hgt.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:Rabs (sum_f_R0 An N0 - l) < l1 - l
H6:l1 <= sum_f_R0 An N0
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  apply H6.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
l1 <= sum_f_R0 An N0
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  unfold l1; apply Rge_le;
    apply (growing_prop (fun k:nat => sum_f_R0 An k)).An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
Un_growing (fun k : nat => sum_f_R0 An k)
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
(N0 >= N)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  apply H1.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
H4:(N0 >= x)%nat
H5:R_dist (sum_f_R0 An N0) l < l1 - l
(N0 >= N)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  unfold ge, N0; apply le_max_r.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
H2:0 < l1 - l
x:nat
H3:forall n : nat,
(n >= x)%nat -> R_dist (sum_f_R0 An n) l < l1 - l
N0:=Nat.max x N:nat
(N0 >= x)%nat
An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  unfold ge, N0; apply le_max_l.An:nat -> R
N:nat
l:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (sum_f_R0 An n) l < eps
H0:forall n : nat, 0 <= An n
l1:=sum_f_R0 An N:R
Hgt:l1 > l
H1:Un_growing (fun n : nat => sum_f_R0 An n)
0 < l1 - l
An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  apply Rplus_lt_reg_l with l; rewrite Rplus_0_r;
    replace (l + (l1 - l)) with l1; [ apply Hgt | ring ].An:nat -> R
N:nat
l:R
H:Un_cv (fun n : nat => sum_f_R0 An n) l
H0:forall n : nat, 0 <= An n
Hgt:sum_f_R0 An N > l
Un_growing (fun n : nat => sum_f_R0 An n)
  unfold Un_growing; intro; simpl;
    pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r;
      apply Rplus_le_compat_l; apply H0.
Qed.

(**********)
Lemma sum_cv_maj :
  forall (An:nat -> R) (fn:nat -> R -> R) (x l1 l2:R),
    Un_cv (fun n:nat => SP fn n x) l1 ->
    Un_cv (fun n:nat => sum_f_R0 An n) l2 ->
    (forall n:nat, Rabs (fn n x) <= An n) -> Rabs l1 <= l2.forall (An : nat -> R) (fn : nat -> R -> R)
  (x l1 l2 : R),
Un_cv (fun n : nat => SP fn n x) l1 ->
Un_cv (fun n : nat => sum_f_R0 An n) l2 ->
(forall n : nat, Rabs (fn n x) <= An n) ->
Rabs l1 <= l2
Proof.forall (An : nat -> R) (fn : nat -> R -> R)
  (x l1 l2 : R),
Un_cv (fun n : nat => SP fn n x) l1 ->
Un_cv (fun n : nat => sum_f_R0 An n) l2 ->
(forall n : nat, Rabs (fn n x) <= An n) ->
Rabs l1 <= l2
  intros; destruct (total_order_T (Rabs l1) l2) as [[Hlt|Heq]|Hgt].An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hlt:Rabs l1 < l2
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Heq:Rabs l1 = l2
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
Rabs l1 <= l2
  left; apply Hlt.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Heq:Rabs l1 = l2
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
Rabs l1 <= l2
  right; apply Heq.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
Rabs l1 <= l2
  cut (forall n0:nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
(forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0) ->
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  intro; cut (0 < (Rabs l1 - l2) / 2).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2 -> Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  intro; unfold Un_cv in H, H0.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  elim (H _ H3); intros Na H4.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
R_dist (SP fn n x) l1 < (Rabs l1 - l2) / 2
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  elim (H0 _ H3); intros Nb H5.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
R_dist (SP fn n x) l1 < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
R_dist (sum_f_R0 An n) l2 < (Rabs l1 - l2) / 2
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  set (N := max Na Nb).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
R_dist (SP fn n x) l1 < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
R_dist (sum_f_R0 An n) l2 < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  unfold R_dist in H4, H5.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  cut (Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2 ->
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  intro; cut (Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2 ->
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  intro; cut (sum_f_R0 An N < (Rabs l1 + l2) / 2).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2 -> Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  intro; cut ((Rabs l1 + l2) / 2 < Rabs (SP fn N x)).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
(Rabs l1 + l2) / 2 < Rabs (SP fn N x) -> Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  intro; cut (sum_f_R0 An N < Rabs (SP fn N x)).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
H9:(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
sum_f_R0 An N < Rabs (SP fn N x) -> Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
H9:(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
sum_f_R0 An N < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  intro; assert (H11 := H2 N).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
H9:(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
H10:sum_f_R0 An N < Rabs (SP fn N x)
H11:Rabs (SP fn N x) <= sum_f_R0 An N
Rabs l1 <= l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
H9:(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
sum_f_R0 An N < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H11 H10)).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
H9:(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
sum_f_R0 An N < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply Rlt_trans with ((Rabs l1 + l2) / 2); assumption.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  destruct (Rcase_abs (Rabs l1 - Rabs (SP fn N x))) as [Hlt|Hge].An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply Rlt_trans with (Rabs l1).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
(Rabs l1 + l2) / 2 < Rabs l1
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
Rabs l1 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply Rmult_lt_reg_l with 2.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
0 < 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
2 * ((Rabs l1 + l2) / 2) < 2 * Rabs l1
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
Rabs l1 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  prove_sup0.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
2 * ((Rabs l1 + l2) / 2) < 2 * Rabs l1
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
Rabs l1 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  unfold Rdiv; rewrite (Rmult_comm 2); rewrite Rmult_assoc;
    rewrite <- Rinv_l_sym.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
(Rabs l1 + l2) * 1 < 2 * Rabs l1
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
2 <> 0
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
Rabs l1 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  rewrite Rmult_1_r; rewrite double; apply Rplus_lt_compat_l; apply Hgt.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
2 <> 0
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
Rabs l1 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  discrR.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hlt:Rabs l1 - Rabs (SP fn N x) < 0
Rabs l1 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply (Rminus_lt _ _ Hlt).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  rewrite (Rabs_right _ Hge) in H7.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs l1 - Rabs (SP fn N x) < (Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
(Rabs l1 + l2) / 2 < Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply Rplus_lt_reg_l with ((Rabs l1 - l2) / 2 - Rabs (SP fn N x)).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs l1 - Rabs (SP fn N x) < (Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
(Rabs l1 - l2) / 2 - Rabs (SP fn N x) +
(Rabs l1 + l2) / 2 <
(Rabs l1 - l2) / 2 - Rabs (SP fn N x) +
Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  replace ((Rabs l1 - l2) / 2 - Rabs (SP fn N x) + (Rabs l1 + l2) / 2) with
  (Rabs l1 - Rabs (SP fn N x)).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs l1 - Rabs (SP fn N x) < (Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
Rabs l1 - Rabs (SP fn N x) <
(Rabs l1 - l2) / 2 - Rabs (SP fn N x) +
Rabs (SP fn N x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs l1 - Rabs (SP fn N x) < (Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
Rabs l1 - Rabs (SP fn N x) =
(Rabs l1 - l2) / 2 - Rabs (SP fn N x) +
(Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l;
    rewrite Rplus_0_r; apply H7.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs l1 - Rabs (SP fn N x) < (Rabs l1 - l2) / 2
H8:sum_f_R0 An N < (Rabs l1 + l2) / 2
Hge:Rabs l1 - Rabs (SP fn N x) >= 0
Rabs l1 - Rabs (SP fn N x) =
(Rabs l1 - l2) / 2 - Rabs (SP fn N x) +
(Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  unfold Rdiv; rewrite Rmult_plus_distr_r;
    rewrite <- (Rmult_comm (/ 2)); rewrite Rmult_minus_distr_l;
      repeat rewrite (Rmult_comm (/ 2)); pattern (Rabs l1) at 1;
        rewrite double_var; unfold Rdiv in |- *; ring.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  destruct (Rcase_abs (sum_f_R0 An N - l2)) as [Hlt|Hge].An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hlt:sum_f_R0 An N - l2 < 0
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply Rlt_trans with l2.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hlt:sum_f_R0 An N - l2 < 0
sum_f_R0 An N < l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hlt:sum_f_R0 An N - l2 < 0
l2 < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply (Rminus_lt _ _ Hlt).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hlt:sum_f_R0 An N - l2 < 0
l2 < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply Rmult_lt_reg_l with 2.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hlt:sum_f_R0 An N - l2 < 0
0 < 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hlt:sum_f_R0 An N - l2 < 0
2 * l2 < 2 * ((Rabs l1 + l2) / 2)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  prove_sup0.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hlt:sum_f_R0 An N - l2 < 0
2 * l2 < 2 * ((Rabs l1 + l2) / 2)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  rewrite (double l2); unfold Rdiv; rewrite (Rmult_comm 2);
    rewrite Rmult_assoc; rewrite <- Rinv_l_sym.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hlt:sum_f_R0 An N - l2 < 0
l2 + l2 < (Rabs l1 + l2) * 1
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hlt:sum_f_R0 An N - l2 < 0
2 <> 0
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  rewrite Rmult_1_r; rewrite (Rplus_comm (Rabs l1)); apply Rplus_lt_compat_l;
    apply Hgt.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hlt:sum_f_R0 An N - l2 < 0
2 <> 0
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  discrR.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
sum_f_R0 An N < (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  rewrite (Rabs_right _ Hge) in H6; apply Rplus_lt_reg_l with (- l2).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:sum_f_R0 An N - l2 < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
- l2 + sum_f_R0 An N < - l2 + (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  replace (- l2 + (Rabs l1 + l2) / 2) with ((Rabs l1 - l2) / 2).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:sum_f_R0 An N - l2 < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
- l2 + sum_f_R0 An N < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:sum_f_R0 An N - l2 < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
(Rabs l1 - l2) / 2 = - l2 + (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  rewrite Rplus_comm; apply H6.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:sum_f_R0 An N - l2 < (Rabs l1 - l2) / 2
H7:Rabs (Rabs l1 - Rabs (SP fn N x)) <
(Rabs l1 - l2) / 2
Hge:sum_f_R0 An N - l2 >= 0
(Rabs l1 - l2) / 2 = - l2 + (Rabs l1 + l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
    rewrite Rmult_minus_distr_l; rewrite Rmult_plus_distr_r;
      pattern l2 at 2; rewrite double_var;
        repeat rewrite (Rmult_comm (/ 2)); rewrite Ropp_plus_distr;
          unfold Rdiv; ring.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply Rle_lt_trans with (Rabs (SP fn N x - l1)).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (Rabs l1 - Rabs (SP fn N x)) <=
Rabs (SP fn N x - l1)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (SP fn N x - l1) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply Rabs_triang_inv2.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
H6:Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
Rabs (SP fn N x - l1) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply H4; unfold ge, N; apply le_max_l.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
H0:forall eps : R,
eps > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
H3:0 < (Rabs l1 - l2) / 2
Na:nat
H4:forall n : nat,
(n >= Na)%nat ->
Rabs (SP fn n x - l1) < (Rabs l1 - l2) / 2
Nb:nat
H5:forall n : nat,
(n >= Nb)%nat ->
Rabs (sum_f_R0 An n - l2) < (Rabs l1 - l2) / 2
N:=Nat.max Na Nb:nat
Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply H5; unfold ge, N; apply le_max_r.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < (Rabs l1 - l2) / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  unfold Rdiv; apply Rmult_lt_0_compat.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < Rabs l1 - l2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply Rplus_lt_reg_l with l2.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
l2 + 0 < l2 + (Rabs l1 - l2)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  rewrite Rplus_0_r; replace (l2 + (Rabs l1 - l2)) with (Rabs l1);
    [ apply Hgt | ring ].An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
H2:forall n0 : nat,
Rabs (SP fn n0 x) <= sum_f_R0 An n0
0 < / 2
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  apply Rinv_0_lt_compat; prove_sup0.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
forall n0 : nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0
  intros; induction  n0 as [| n0 Hrecn0].An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
Rabs (SP fn 0 x) <= sum_f_R0 An 0
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
Rabs (SP fn (S n0) x) <= sum_f_R0 An (S n0)
  unfold SP; simpl; apply H1.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
Rabs (SP fn (S n0) x) <= sum_f_R0 An (S n0)
  unfold SP; simpl.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
Rabs
  (sum_f_R0 (fun k : nat => fn k x) n0 + fn (S n0) x) <=
sum_f_R0 An n0 + An (S n0)
  apply Rle_trans with
    (Rabs (sum_f_R0 (fun k:nat => fn k x) n0) + Rabs (fn (S n0) x)).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
Rabs
  (sum_f_R0 (fun k : nat => fn k x) n0 + fn (S n0) x) <=
Rabs (sum_f_R0 (fun k : nat => fn k x) n0) +
Rabs (fn (S n0) x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
Rabs (sum_f_R0 (fun k : nat => fn k x) n0) +
Rabs (fn (S n0) x) <= sum_f_R0 An n0 + An (S n0)
  apply Rabs_triang.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
Rabs (sum_f_R0 (fun k : nat => fn k x) n0) +
Rabs (fn (S n0) x) <= sum_f_R0 An n0 + An (S n0)
  apply Rle_trans with (sum_f_R0 An n0 + Rabs (fn (S n0) x)).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
Rabs (sum_f_R0 (fun k : nat => fn k x) n0) +
Rabs (fn (S n0) x) <=
sum_f_R0 An n0 + Rabs (fn (S n0) x)
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
sum_f_R0 An n0 + Rabs (fn (S n0) x) <=
sum_f_R0 An n0 + An (S n0)
  do 2 rewrite <- (Rplus_comm (Rabs (fn (S n0) x))).An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
Rabs (fn (S n0) x) +
Rabs (sum_f_R0 (fun k : nat => fn k x) n0) <=
Rabs (fn (S n0) x) + sum_f_R0 An n0
An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
sum_f_R0 An n0 + Rabs (fn (S n0) x) <=
sum_f_R0 An n0 + An (S n0)
  apply Rplus_le_compat_l; apply Hrecn0.An:nat -> R
fn:nat -> R -> R
x, l1, l2:R
H:Un_cv (fun n : nat => SP fn n x) l1
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
Hgt:Rabs l1 > l2
n0:nat
Hrecn0:Rabs (SP fn n0 x) <= sum_f_R0 An n0
sum_f_R0 An n0 + Rabs (fn (S n0) x) <=
sum_f_R0 An n0 + An (S n0)
  apply Rplus_le_compat_l; apply H1.
Qed.