SeqSeries.v

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(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

Require Import Rbase.
Require Import Rfunctions.
Require Import Max.
Require Export Rseries.
Require Export SeqProp.
Require Export Rcomplete.
Require Export PartSum.
Require Export AltSeries.
Require Export Binomial.
Require Export Rsigma.
Require Export Rprod.
Require Export Cauchy_prod.
Require Export Alembert.
Local Open Scope R_scope.

(**********)
Lemma sum_maj1 :
  forall (fn:nat -> R -> R) (An:nat -> R) (x l1 l2:R)
    (N:nat),
    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 - SP fn N x) <= l2 - sum_f_R0 An N.forall (fn : nat -> R -> R) (An : nat -> R)
  (x l1 l2 : R) (N : nat),
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 - SP fn N x) <= l2 - sum_f_R0 An N
Proof.forall (fn : nat -> R -> R) (An : nat -> R)
  (x l1 l2 : R) (N : nat),
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 - SP fn N x) <= l2 - sum_f_R0 An N
  intros;
    cut { l:R | Un_cv (fun n => sum_f_R0 (fun l => fn (S N + l)%nat x) n) l }.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l} ->
Rabs (l1 - SP fn N x) <= l2 - sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intro X;
    cut { l:R | Un_cv (fun n => sum_f_R0 (fun l => An (S N + l)%nat) n) l }.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l} ->
Rabs (l1 - SP fn N x) <= l2 - sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intro X0; elim X; intros l1N H2.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
Rabs (l1 - SP fn N x) <= l2 - sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  elim X0; intros l2N H3.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
Rabs (l1 - SP fn N x) <= l2 - sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  cut (l1 - SP fn N x = l1N).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N ->
Rabs (l1 - SP fn N x) <= l2 - sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intro; cut (l2 - sum_f_R0 An N = l2N).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
l2 - sum_f_R0 An N = l2N ->
Rabs (l1 - SP fn N x) <= l2 - sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
l2 - sum_f_R0 An N = l2N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intro; rewrite H4; rewrite H5.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
H5:l2 - sum_f_R0 An N = l2N
Rabs l1N <= l2N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
l2 - sum_f_R0 An N = l2N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply sum_cv_maj with
    (fun l:nat => An (S N + l)%nat) (fun (l:nat) (x:R) => fn (S N + l)%nat x) x.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
H5:l2 - sum_f_R0 An N = l2N
Un_cv
  (fun n : nat =>
   SP (fun (l : nat) (x0 : R) => fn (S N + l)%nat x0)
     n x) l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
H5:l2 - sum_f_R0 An N = l2N
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n) l2N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
H5:l2 - sum_f_R0 An N = l2N
forall n : nat,
Rabs (fn (S N + n)%nat x) <= An (S N + n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
l2 - sum_f_R0 An N = l2N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold SP; apply H2.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
H5:l2 - sum_f_R0 An N = l2N
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n) l2N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
H5:l2 - sum_f_R0 An N = l2N
forall n : nat,
Rabs (fn (S N + n)%nat x) <= An (S N + n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
l2 - sum_f_R0 An N = l2N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H3.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
H5:l2 - sum_f_R0 An N = l2N
forall n : nat,
Rabs (fn (S N + n)%nat x) <= An (S N + n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
l2 - sum_f_R0 An N = l2N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intros; apply H1.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
l2 - sum_f_R0 An N = l2N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  symmetry ; eapply UL_sequence.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
Un_cv ?Un l2N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
Un_cv ?Un (l2 - sum_f_R0 An N)
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H3.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  (l2 - sum_f_R0 An N)
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold Un_cv in H0; unfold Un_cv; intros; elim (H0 eps H5);
    intros N0 H6.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n : nat => SP fn n x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps0
H1:forall n : nat, Rabs (fn n x) <= An n
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n : nat,
(n >= N0)%nat -> R_dist (sum_f_R0 An n) l2 < eps
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat ->
  R_dist
    (sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
    (l2 - sum_f_R0 An N) < eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold R_dist in H6; exists N0; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
R_dist (sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  (l2 - sum_f_R0 An N) < eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold R_dist;
    replace (sum_f_R0 (fun l:nat => An (S N + l)%nat) n - (l2 - sum_f_R0 An N))
    with (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n - l2);
      [ idtac | ring ].fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
Rabs
  (sum_f_R0 An N +
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n - l2) <
eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  replace (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n) with
  (sum_f_R0 An (S (N + n))).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
Rabs (sum_f_R0 An (S (N + n)) - l2) < eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H6; unfold ge; apply le_trans with n.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N0 <= n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H7.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_trans with (N + n)%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(n <= N + n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N + n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_plus_r.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N + n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_n_Sn.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  cut (0 <= N)%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat ->
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  cut (N < S (N + n))%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat ->
(0 <= N)%nat ->
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intros; assert (H10 := sigma_split An H9 H8).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sigma An 0 (S (N + n)) = sigma An 0 N + sigma An (S N) (S (N + n))
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold sigma in H10.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n) - 0) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) (N - 0) +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  do 2 rewrite <- minus_n_O in H10.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  replace (sum_f_R0 An (S (N + n))) with
  (sum_f_R0 (fun k:nat => An (0 + k)%nat) (S (N + n))).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  replace (sum_f_R0 An N) with (sum_f_R0 (fun k:nat => An (0 + k)%nat) N).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  cut ((S (N + n) - S N)%nat = n).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
(S (N + n) - S N)%nat = n ->
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intro; rewrite H11 in H10.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) n
H11:(S (N + n) - S N)%nat = n
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H10.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply INR_eq; rewrite minus_INR.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
INR (S (N + n)) - INR (S N) = INR n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
(S N <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  do 2 rewrite S_INR; rewrite plus_INR; ring.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
(S N <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_n_S; apply le_plus_l.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply sum_eq; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
i:nat
H11:(i <= N)%nat
An (0 + i)%nat = An i
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  reflexivity.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply sum_eq; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
H8:(N < S (N + n))%nat
H9:(0 <= N)%nat
H10:sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) (S (N + n) - S N)
i:nat
H11:(i <= S (N + n))%nat
An (0 + i)%nat = An i
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  reflexivity.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_lt_n_Sm; apply le_plus_l.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
H4:l1 - SP fn N x = l1N
eps:R
H5:eps > 0
N0:nat
H6:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H7:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_O_n.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
l1 - SP fn N x = l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  symmetry ; eapply UL_sequence.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
Un_cv ?Un l1N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
Un_cv ?Un (l1 - SP fn N x)
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H2.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  (l1 - SP fn N x)
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold Un_cv in H; unfold Un_cv; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps0
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
eps:R
H4:eps > 0
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist
    (sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
    (l1 - SP fn N x) < eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  elim (H eps H4); intros N0 H5.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (SP fn n x) l1 < eps0
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
X0:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n)
  l}
l1N:R
H2:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  l1N
l2N:R
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n : nat,
(n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat ->
  R_dist
    (sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
    (l1 - SP fn N x) < eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold R_dist in H5; exists N0; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
R_dist
  (sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  (l1 - SP fn N x) < eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold R_dist, SP;
    replace
    (sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n -
      (l1 - sum_f_R0 (fun k:nat => fn k x) N)) with
    (sum_f_R0 (fun k:nat => fn k x) N +
      sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - l1);
    [ idtac | ring ].fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
Rabs
  (sum_f_R0 (fun k : nat => fn k x) N +
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n - l1) <
eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  replace
  (sum_f_R0 (fun k:nat => fn k x) N +
    sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) with
  (sum_f_R0 (fun k:nat => fn k x) (S (N + n))).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
Rabs
  (sum_f_R0 (fun k : nat => fn k x) (S (N + n)) - l1) <
eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold SP in H5; apply H5; unfold ge; apply le_trans with n.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H6:(n >= N0)%nat
(N0 <= n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H6:(n >= N0)%nat
(n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H6.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H6:(n >= N0)%nat
(n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_trans with (N + n)%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H6:(n >= N0)%nat
(n <= N + n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H6:(n >= N0)%nat
(N + n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_plus_r.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H6:(n >= N0)%nat
(N + n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_n_Sn.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  cut (0 <= N)%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat ->
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  cut (N < S (N + n))%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat ->
(0 <= N)%nat ->
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intros; assert (H9 := sigma_split (fun k:nat => fn k x) H8 H7).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sigma (fun k : nat => fn k x) 0 (S (N + n)) =
sigma (fun k : nat => fn k x) 0 N +
sigma (fun k : nat => fn k x) (S N) (S (N + n))
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold sigma in H9.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n) - 0) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (N - 0) +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  do 2 rewrite <- minus_n_O in H9.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  replace (sum_f_R0 (fun k:nat => fn k x) (S (N + n))) with
  (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) (S (N + n))).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  replace (sum_f_R0 (fun k:nat => fn k x) N) with
  (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) N).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  cut ((S (N + n) - S N)%nat = n).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n ->
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intro; rewrite H10 in H9.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x) n
H10:(S (N + n) - S N)%nat = n
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H9.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply INR_eq; rewrite minus_INR.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
INR (S (N + n)) - INR (S N) = INR n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S N <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  do 2 rewrite S_INR; rewrite plus_INR; ring.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S N <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_n_S; apply le_plus_l.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply sum_eq; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
i:nat
H10:(i <= N)%nat
fn (0 + i)%nat x = fn i x
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  reflexivity.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply sum_eq; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
H7:(N < S (N + n))%nat
H8:(0 <= N)%nat
H9:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
i:nat
H10:(i <= S (N + n))%nat
fn (0 + i)%nat x = fn i x
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  reflexivity.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_lt_n_Sm.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(N <= N + n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_plus_l.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
X0:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat)
     n0) l}
l1N:R
H2:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x)
     n0) l1N
l2N:R
H3:Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n0)
  l2N
eps:R
H4:eps > 0
N0:nat
H5:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H6:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_O_n.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => An (S N + l0)%nat) n) l}
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  exists (l2 - sum_f_R0 An N).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  (l2 - sum_f_R0 An N)
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold Un_cv in H0; unfold Un_cv; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n : nat => SP fn n x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps0
H1:forall n : nat, Rabs (fn n x) <= An n
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
eps:R
H2:eps > 0
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist
    (sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
    (l2 - sum_f_R0 An N) < eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  elim (H0 eps H2); intros N0 H3.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n : nat => SP fn n x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat ->
  R_dist (sum_f_R0 An n) l2 < eps0
H1:forall n : nat, Rabs (fn n x) <= An n
X:{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n : nat,
(n >= N0)%nat -> R_dist (sum_f_R0 An n) l2 < eps
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat ->
  R_dist
    (sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
    (l2 - sum_f_R0 An N) < eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold R_dist in H3; exists N0; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
R_dist (sum_f_R0 (fun l : nat => An (S N + l)%nat) n)
  (l2 - sum_f_R0 An N) < eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold R_dist;
    replace (sum_f_R0 (fun l:nat => An (S N + l)%nat) n - (l2 - sum_f_R0 An N))
    with (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n - l2);
      [ idtac | ring ].fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
Rabs
  (sum_f_R0 An N +
   sum_f_R0 (fun l : nat => An (S N + l)%nat) n - l2) <
eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  replace (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n) with
  (sum_f_R0 An (S (N + n))).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
Rabs (sum_f_R0 An (S (N + n)) - l2) < eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H3; unfold ge; apply le_trans with n.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N0 <= n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H4.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_trans with (N + n)%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(n <= N + n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N + n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_plus_r.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N + n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_n_Sn.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  cut (0 <= N)%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat ->
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  cut (N < S (N + n))%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat ->
(0 <= N)%nat ->
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intros; assert (H7 := sigma_split An H6 H5).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sigma An 0 (S (N + n)) =
sigma An 0 N + sigma An (S N) (S (N + n))
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  unfold sigma in H7.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n) - 0) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) (N - 0) +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  do 2 rewrite <- minus_n_O in H7.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 An (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  replace (sum_f_R0 An (S (N + n))) with
  (sum_f_R0 (fun k:nat => An (0 + k)%nat) (S (N + n))).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  replace (sum_f_R0 An N) with (sum_f_R0 (fun k:nat => An (0 + k)%nat) N).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  cut ((S (N + n) - S N)%nat = n).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n ->
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  intro; rewrite H8 in H7.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat) n
H8:(S (N + n) - S N)%nat = n
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun l : nat => An (S N + l)%nat) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply H7.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply INR_eq; rewrite minus_INR.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
INR (S (N + n)) - INR (S N) = INR n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
(S N <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  do 2 rewrite S_INR; rewrite plus_INR; ring.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
(S N <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_n_S; apply le_plus_l.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) N =
sum_f_R0 An N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply sum_eq; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
i:nat
H8:(i <= N)%nat
An (0 + i)%nat = An i
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  reflexivity.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => An (0 + k)%nat) (S (N + n)) =
sum_f_R0 An (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply sum_eq; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => An (0 + k)%nat)
  (S (N + n)) =
sum_f_R0 (fun k : nat => An (0 + k)%nat) N +
sum_f_R0 (fun k : nat => An (S N + k)%nat)
  (S (N + n) - S N)
i:nat
H8:(i <= S (N + n))%nat
An (0 + i)%nat = An i
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  reflexivity.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_lt_n_Sm.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(N <= N + n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_plus_l.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:Un_cv (fun n0 : nat => SP fn n0 x) l1
H0:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat ->
  R_dist (sum_f_R0 An n0) l2 < eps0
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
X:{l : R |
Un_cv
  (fun n0 : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x)
     n0) l}
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 An n0 - l2) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  apply le_O_n.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l0 : nat => fn (S N + l0)%nat x) n) l}
  exists (l1 - SP fn N x).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
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
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  (l1 - SP fn N x)
  unfold Un_cv in H; unfold Un_cv; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (SP fn n x) l1 < eps0
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
eps:R
H2:eps > 0
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  R_dist
    (sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
    (l1 - SP fn N x) < eps
  elim (H eps H2); intros N0 H3.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat -> R_dist (SP fn n x) l1 < eps0
H0:Un_cv (fun n : nat => sum_f_R0 An n) l2
H1:forall n : nat, Rabs (fn n x) <= An n
eps:R
H2:eps > 0
N0:nat
H3:forall n : nat,
(n >= N0)%nat -> R_dist (SP fn n x) l1 < eps
exists N1 : nat,
  forall n : nat,
  (n >= N1)%nat ->
  R_dist
    (sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
    (l1 - SP fn N x) < eps
  unfold R_dist in H3; exists N0; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
R_dist
  (sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n)
  (l1 - SP fn N x) < eps
  unfold R_dist, SP.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
Rabs
  (sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n -
   (l1 - sum_f_R0 (fun k : nat => fn k x) N)) < eps
  replace
  (sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n -
    (l1 - sum_f_R0 (fun k:nat => fn k x) N)) with
  (sum_f_R0 (fun k:nat => fn k x) N +
    sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - l1);
  [ idtac | ring ].fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
Rabs
  (sum_f_R0 (fun k : nat => fn k x) N +
   sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n - l1) <
eps
  replace
  (sum_f_R0 (fun k:nat => fn k x) N +
    sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) with
  (sum_f_R0 (fun k:nat => fn k x) (S (N + n))).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
Rabs
  (sum_f_R0 (fun k : nat => fn k x) (S (N + n)) - l1) <
eps
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
  unfold SP in H3; apply H3.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H4:(n >= N0)%nat
(S (N + n) >= N0)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
  unfold ge; apply le_trans with n.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H4:(n >= N0)%nat
(N0 <= n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H4:(n >= N0)%nat
(n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
  apply H4.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H4:(n >= N0)%nat
(n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
  apply le_trans with (N + n)%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H4:(n >= N0)%nat
(n <= N + n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H4:(n >= N0)%nat
(N + n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
  apply le_plus_r.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (sum_f_R0 (fun k : nat => fn k x) n0 - l1) <
eps
n:nat
H4:(n >= N0)%nat
(N + n <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
  apply le_n_Sn.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
  cut (0 <= N)%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat ->
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  cut (N < S (N + n))%nat.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat ->
(0 <= N)%nat ->
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  intros; assert (H7 := sigma_split (fun k:nat => fn k x) H6 H5).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sigma (fun k : nat => fn k x) 0 (S (N + n)) =
sigma (fun k : nat => fn k x) 0 N +
sigma (fun k : nat => fn k x) (S N) (S (N + n))
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  unfold sigma in H7.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n) - 0) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (N - 0) +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  do 2 rewrite <- minus_n_O in H7.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn k x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  replace (sum_f_R0 (fun k:nat => fn k x) (S (N + n))) with
  (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) (S (N + n))).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  replace (sum_f_R0 (fun k:nat => fn k x) N) with
  (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) N).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  cut ((S (N + n) - S N)%nat = n).fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n ->
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  intro; rewrite H8 in H7.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x) n
H8:(S (N + n) - S N)%nat = n
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun l : nat => fn (S N + l)%nat x) n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  apply H7.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S (N + n) - S N)%nat = n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  apply INR_eq; rewrite minus_INR.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
INR (S (N + n)) - INR (S N) = INR n
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S N <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  do 2 rewrite S_INR; rewrite plus_INR; ring.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
(S N <= S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  apply le_n_S; apply le_plus_l.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N =
sum_f_R0 (fun k : nat => fn k x) N
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  apply sum_eq; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
i:nat
H8:(i <= N)%nat
fn (0 + i)%nat x = fn i x
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  reflexivity.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) (S (N + n)) =
sum_f_R0 (fun k : nat => fn k x) (S (N + n))
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  apply sum_eq; intros.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
H5:(N < S (N + n))%nat
H6:(0 <= N)%nat
H7:sum_f_R0 (fun k : nat => fn (0 + k)%nat x)
  (S (N + n)) =
sum_f_R0 (fun k : nat => fn (0 + k)%nat x) N +
sum_f_R0 (fun k : nat => fn (S N + k)%nat x)
  (S (N + n) - S N)
i:nat
H8:(i <= S (N + n))%nat
fn (0 + i)%nat x = fn i x
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  reflexivity.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N < S (N + n))%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  apply le_lt_n_Sm.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(N <= N + n)%nat
fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  apply le_plus_l.fn:nat -> R -> R
An:nat -> R
x, l1, l2:R
N:nat
H:forall eps0 : R,
eps0 > 0 ->
exists N1 : nat,
  forall n0 : nat,
  (n0 >= N1)%nat -> R_dist (SP fn n0 x) l1 < eps0
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) l2
H1:forall n0 : nat, Rabs (fn n0 x) <= An n0
eps:R
H2:eps > 0
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (SP fn n0 x - l1) < eps
n:nat
H4:(n >= N0)%nat
(0 <= N)%nat
  apply le_O_n.
Qed.

Comparaison of convergence for series

Lemma Rseries_CV_comp :
  forall An Bn:nat -> R,
    (forall n:nat, 0 <= An n <= Bn n) ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 Bn N) l } ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.forall An Bn : nat -> R,
(forall n : nat, 0 <= An n <= Bn n) ->
{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l} ->
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
Proof.forall An Bn : nat -> R,
(forall n : nat, 0 <= An n <= Bn n) ->
{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l} ->
{l : R | Un_cv (fun N : nat => sum_f_R0 An N) l}
  intros An Bn H X; apply cv_cauchy_2.An, Bn:nat -> R
H:forall n : nat, 0 <= An n <= Bn n
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
Cauchy_crit_series An
  assert (H0 := cv_cauchy_1 _ X).An, Bn:nat -> R
H:forall n : nat, 0 <= An n <= Bn n
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
Cauchy_crit_series An
  unfold Cauchy_crit_series; unfold Cauchy_crit.An, Bn:nat -> R
H:forall n : nat, 0 <= An n <= Bn n
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
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; elim (H0 eps H1); intros.An, Bn:nat -> R
H:forall n : nat, 0 <= An n <= Bn n
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n m : nat,
(n >= x)%nat ->
(m >= x)%nat ->
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn 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; intros.An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(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 Bn n) (sum_f_R0 Bn m)).An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m) ->
R_dist (sum_f_R0 An n) (sum_f_R0 An m) < eps
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)
  intro; apply Rle_lt_trans with (R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)).An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
H5:R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
H5:R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m) < eps
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)
  assumption.An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
H5:R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m) < eps
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)
  apply H2; assumption.An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)
  destruct (lt_eq_lt_dec n m) as [[| -> ]|].An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)
An, Bn:nat -> R
H:forall n : nat, 0 <= An n <= Bn n
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m0) < eps
m:nat
H3, H4:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn m) (sum_f_R0 Bn m)
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)
  -An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m) rewrite (tech2 An n m); [ idtac | assumption ].An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(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 Bn n) (sum_f_R0 Bn m)
    rewrite (tech2 Bn n m); [ idtac | assumption ].An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(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 Bn n)
  (sum_f_R0 Bn n +
   sum_f_R0 (fun i : nat => Bn (S n + i)%nat)
     (m - S n))
    unfold R_dist; unfold Rminus; do 2 rewrite Ropp_plus_distr;
      do 2 rewrite <- Rplus_assoc; do 2 rewrite Rplus_opp_r;
        do 2 rewrite Rplus_0_l; do 2 rewrite Rabs_Ropp; repeat rewrite Rabs_right.An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => An (S n + i)%nat) (m - S n) <=
sum_f_R0 (fun i : nat => Bn (S n + i)%nat) (m - S n)
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => Bn (S n + i)%nat) (m - S n) >=
0
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => An (S n + i)%nat) (m - S n) >=
0
    apply sum_Rle; intros.An, Bn:nat -> R
H:forall n1 : nat, 0 <= An n1 <= Bn n1
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n1 m0 : nat,
(n1 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n1) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
n0:nat
H5:(n0 <= m - S n)%nat
An (S n + n0)%nat <= Bn (S n + n0)%nat
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => Bn (S n + i)%nat) (m - S n) >=
0
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => An (S n + i)%nat) (m - S n) >=
0
    elim (H (S n + n0)%nat); intros H7 H8.An, Bn:nat -> R
H:forall n1 : nat, 0 <= An n1 <= Bn n1
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n1 m0 : nat,
(n1 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n1) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
n0:nat
H5:(n0 <= m - S n)%nat
H7:0 <= An (S n + n0)%nat
H8:An (S n + n0)%nat <= Bn (S n + n0)%nat
An (S n + n0)%nat <= Bn (S n + n0)%nat
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => Bn (S n + i)%nat) (m - S n) >=
0
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => An (S n + i)%nat) (m - S n) >=
0
    apply H8.An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => Bn (S n + i)%nat) (m - S n) >=
0
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => An (S n + i)%nat) (m - S n) >=
0
    apply Rle_ge; apply cond_pos_sum; intro.An, Bn:nat -> R
H:forall n1 : nat, 0 <= An n1 <= Bn n1
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n1 m0 : nat,
(n1 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n1) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
n0:nat
0 <= Bn (S n + n0)%nat
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => An (S n + i)%nat) (m - S n) >=
0
    elim (H (S n + n0)%nat); intros.An, Bn:nat -> R
H:forall n1 : nat, 0 <= An n1 <= Bn n1
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n1 m0 : nat,
(n1 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n1) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
n0:nat
H5:0 <= An (S n + n0)%nat
H6:An (S n + n0)%nat <= Bn (S n + n0)%nat
0 <= Bn (S n + n0)%nat
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => An (S n + i)%nat) (m - S n) >=
0
    apply Rle_trans with (An (S n + n0)%nat); assumption.An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
sum_f_R0 (fun i : nat => An (S n + i)%nat) (m - S n) >=
0
    apply Rle_ge; apply cond_pos_sum; intro.An, Bn:nat -> R
H:forall n1 : nat, 0 <= An n1 <= Bn n1
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n1 m0 : nat,
(n1 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n1) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(n < m)%nat
n0:nat
0 <= An (S n + n0)%nat
    elim (H (S n + n0)%nat); intros; assumption.
  -An, Bn:nat -> R
H:forall n : nat, 0 <= An n <= Bn n
X:{l : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n m0 : nat,
(n >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m0) < eps
m:nat
H3, H4:(m >= x)%nat
R_dist (sum_f_R0 An m) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn m) (sum_f_R0 Bn m) unfold R_dist; unfold Rminus;
    do 2 rewrite Rplus_opp_r; rewrite Rabs_R0; right;
      reflexivity.
  -An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
R_dist (sum_f_R0 An n) (sum_f_R0 An m) <=
R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m) rewrite (tech2 An m n); [ idtac | assumption ].An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(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 Bn n) (sum_f_R0 Bn m)
    rewrite (tech2 Bn m n); [ idtac | assumption ].An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(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 Bn m +
   sum_f_R0 (fun i : nat => Bn (S m + i)%nat)
     (n - S m)) (sum_f_R0 Bn m)
    unfold R_dist; unfold Rminus; do 2 rewrite Rplus_assoc;
      rewrite (Rplus_comm (sum_f_R0 An m)); rewrite (Rplus_comm (sum_f_R0 Bn m));
        do 2 rewrite Rplus_assoc; do 2 rewrite Rplus_opp_l;
          do 2 rewrite Rplus_0_r; repeat rewrite Rabs_right.An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) <=
sum_f_R0 (fun i : nat => Bn (S m + i)%nat) (n - S m)
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => Bn (S m + i)%nat) (n - S m) >=
0
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) >=
0
    apply sum_Rle; intros.An, Bn:nat -> R
H:forall n1 : nat, 0 <= An n1 <= Bn n1
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n1 m0 : nat,
(n1 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n1) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
n0:nat
H5:(n0 <= n - S m)%nat
An (S m + n0)%nat <= Bn (S m + n0)%nat
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => Bn (S m + i)%nat) (n - S m) >=
0
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) >=
0
    elim (H (S m + n0)%nat); intros H7 H8; apply H8.An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => Bn (S m + i)%nat) (n - S m) >=
0
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) >=
0
    apply Rle_ge; apply cond_pos_sum; intro.An, Bn:nat -> R
H:forall n1 : nat, 0 <= An n1 <= Bn n1
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n1 m0 : nat,
(n1 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n1) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
n0:nat
0 <= Bn (S m + n0)%nat
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) >=
0
    elim (H (S m + n0)%nat); intros.An, Bn:nat -> R
H:forall n1 : nat, 0 <= An n1 <= Bn n1
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n1 m0 : nat,
(n1 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n1) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
n0:nat
H5:0 <= An (S m + n0)%nat
H6:An (S m + n0)%nat <= Bn (S m + n0)%nat
0 <= Bn (S m + n0)%nat
An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) >=
0
    apply Rle_trans with (An (S m + n0)%nat); assumption.An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m) >=
0
    apply Rle_ge.An, Bn:nat -> R
H:forall n0 : nat, 0 <= An n0 <= Bn n0
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n0 m0 : nat,
(n0 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n0) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
0 <=
sum_f_R0 (fun i : nat => An (S m + i)%nat) (n - S m)
    apply cond_pos_sum; intro.An, Bn:nat -> R
H:forall n1 : nat, 0 <= An n1 <= Bn n1
X:{l0 : R | Un_cv (fun N : nat => sum_f_R0 Bn N) l0}
H0:Cauchy_crit_series Bn
eps:R
H1:eps > 0
x:nat
H2:forall n1 m0 : nat,
(n1 >= x)%nat ->
(m0 >= x)%nat ->
R_dist (sum_f_R0 Bn n1) (sum_f_R0 Bn m0) < eps
n, m:nat
H3:(n >= x)%nat
H4:(m >= x)%nat
l:(m < n)%nat
n0:nat
0 <= An (S m + n0)%nat
    elim (H (S m + n0)%nat); intros; assumption.
Qed.

Cesaro's theorem

Lemma Cesaro :
  forall (An Bn:nat -> R) (l:R),
    Un_cv Bn l ->
    (forall n:nat, 0 < An n) ->
    cv_infty (fun n:nat => sum_f_R0 An n) ->
    Un_cv (fun n:nat => sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n)
    l.forall (An Bn : nat -> R) (l : R),
Un_cv Bn l ->
(forall n : nat, 0 < An n) ->
cv_infty (fun n : nat => sum_f_R0 An n) ->
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun k : nat => An k * Bn k) n /
   sum_f_R0 An n) l
Proof with trivial.forall (An Bn : nat -> R) (l : R),
Un_cv Bn l ->
(forall n : nat, 0 < An n) ->
cv_infty (fun n : nat => sum_f_R0 An n) ->
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun k : nat => An k * Bn k) n /
   sum_f_R0 An n) l
  unfold Un_cv; intros; assert (H3 : forall n:nat, 0 < sum_f_R0 An n)...An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
forall n : nat, 0 < sum_f_R0 An n
An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  intro; apply tech1...An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  assert (H4 : forall n:nat, sum_f_R0 An n <> 0)...An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
forall n : nat, sum_f_R0 An n <> 0
An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  intro; red; intro; assert (H5 := H3 n); rewrite H4 in H5;
    elim (Rlt_irrefl _ H5)...An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  assert (H5 := cv_infty_cv_R0 _ H4 H1); assert (H6 : 0 < eps / 2)...An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
0 < eps / 2
An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  unfold Rdiv; apply Rmult_lt_0_compat...An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
0 < / 2
An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  apply Rinv_0_lt_compat; prove_sup...An, Bn:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Bn n) l < eps0
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  elim (H _ H6); clear H; intros N1 H;
    set (C := Rabs (sum_f_R0 (fun k:nat => An k * (Bn k - l)) N1));
      assert
        (H7 :
          exists N : nat,
            (forall n:nat, (N <= n)%nat -> C / sum_f_R0 An n < eps / 2))...An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  case (Req_dec C 0); intro...An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C = 0
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  exists 0%nat; intros...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C = 0
n:nat
H8:(0 <= n)%nat
C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  rewrite H7; unfold Rdiv; rewrite Rmult_0_l; apply Rmult_lt_0_compat...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C = 0
n:nat
H8:(0 <= n)%nat
0 < / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  apply Rinv_0_lt_compat; prove_sup...An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  assert (H8 : 0 < eps / (2 * Rabs C))...An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
0 < eps / (2 * Rabs C)
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  unfold Rdiv; apply Rmult_lt_0_compat...An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
0 < / (2 * Rabs C)
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  apply Rinv_0_lt_compat; apply Rmult_lt_0_compat...An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
0 < 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
0 < Rabs C
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  prove_sup...An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
0 < Rabs C
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  apply Rabs_pos_lt...An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  elim (H5 _ H8); intros; exists x; intros; assert (H11 := H9 _ H10);
    unfold R_dist in H11; unfold Rminus in H11; rewrite Ropp_0 in H11;
      rewrite Rplus_0_r in H11...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
C / sum_f_R0 An n < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  apply Rle_lt_trans with (Rabs (C / sum_f_R0 An n))...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
C / sum_f_R0 An n <= Rabs (C / sum_f_R0 An n)
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs (C / sum_f_R0 An n) < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  apply RRle_abs...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs (C / sum_f_R0 An n) < eps / 2
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  unfold Rdiv; rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs C)...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
0 < / Rabs C
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
/ Rabs C * (Rabs C * Rabs (/ sum_f_R0 An n)) <
/ Rabs C * (eps * / 2)
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  apply Rinv_0_lt_compat; apply Rabs_pos_lt...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
/ Rabs C * (Rabs C * Rabs (/ sum_f_R0 An n)) <
/ Rabs C * (eps * / 2)
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
1 * Rabs (/ sum_f_R0 An n) < / Rabs C * (eps * / 2)
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs C <> 0
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  rewrite Rmult_1_l; replace (/ Rabs C * (eps * / 2)) with (eps / (2 * Rabs C))...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
eps / (2 * Rabs C) = / Rabs C * (eps * / 2)
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs C <> 0
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  unfold Rdiv; rewrite Rinv_mult_distr...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
eps * (/ 2 * / Rabs C) = / Rabs C * (eps * / 2)
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
2 <> 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs C <> 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs C <> 0
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  ring...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
2 <> 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs C <> 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs C <> 0
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  discrR...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs C <> 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs C <> 0
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  apply Rabs_no_R0...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:C <> 0
H8:0 < eps / (2 * Rabs C)
x:nat
H9:forall n0 : nat,
(n0 >= x)%nat ->
R_dist (/ sum_f_R0 An n0) 0 < eps / (2 * Rabs C)
n:nat
H10:(x <= n)%nat
H11:Rabs (/ sum_f_R0 An n) < eps / (2 * Rabs C)
Rabs C <> 0
An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  apply Rabs_no_R0...An, Bn:nat -> R
l:R
H0:forall n : nat, 0 < An n
H1:cv_infty (fun n : nat => sum_f_R0 An n)
eps:R
H2:eps > 0
H3:forall n : nat, 0 < sum_f_R0 An n
H4:forall n : nat, sum_f_R0 An n <> 0
H5:Un_cv (fun n : nat => / sum_f_R0 An n) 0
H6:0 < eps / 2
N1:nat
H:forall n : nat,
(n >= N1)%nat -> R_dist (Bn n) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
H7:exists N : nat,
  forall n : nat,
  (N <= n)%nat -> C / sum_f_R0 An n < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n /
     sum_f_R0 An n) l < eps
  elim H7; clear H7; intros N2 H7; set (N := max N1 N2); exists (S N); intros;
    unfold R_dist;
      replace (sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n - l) with
      (sum_f_R0 (fun k:nat => An k * (Bn k - l)) n / sum_f_R0 An n)...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
   sum_f_R0 An n) < eps
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  assert (H9 : (N1 < n)%nat)...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
(N1 < n)%nat
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
   sum_f_R0 An n) < eps
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply lt_le_trans with (S N)...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
(N1 < S N)%nat
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
   sum_f_R0 An n) < eps
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply le_lt_n_Sm; unfold N; apply le_max_l...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
   sum_f_R0 An n) < eps
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  rewrite (tech2 (fun k:nat => An k * (Bn k - l)) _ _ H9); unfold Rdiv;
    rewrite Rmult_plus_distr_r;
      apply Rle_lt_trans with
        (Rabs (sum_f_R0 (fun k:nat => An k * (Bn k - l)) N1 / sum_f_R0 An n) +
          Rabs
          (sum_f_R0 (fun i:nat => An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
            (n - S N1) / sum_f_R0 An n))...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1 *
   / sum_f_R0 An n +
   sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1) * / sum_f_R0 An n) <=
Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1 /
   sum_f_R0 An n) +
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1) / sum_f_R0 An n)
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1 /
   sum_f_R0 An n) +
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1) / sum_f_R0 An n) < eps
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply Rabs_triang...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1 /
   sum_f_R0 An n) +
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1) / sum_f_R0 An n) < eps
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  rewrite (double_var eps); apply Rplus_lt_compat...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1 /
   sum_f_R0 An n) < eps / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1) / sum_f_R0 An n) < eps / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  unfold Rdiv; rewrite Rabs_mult; fold C; rewrite Rabs_right...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
C * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1) / sum_f_R0 An n) < eps / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply (H7 n); apply le_trans with (S N)...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
(N2 <= S N)%nat
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1) / sum_f_R0 An n) < eps / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply le_trans with N; [ unfold N; apply le_max_r | apply le_n_Sn ]...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1) / sum_f_R0 An n) < eps / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply Rle_ge; left; apply Rinv_0_lt_compat...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1) / sum_f_R0 An n) < eps / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l

  unfold R_dist in H; unfold Rdiv; rewrite Rabs_mult;
    rewrite (Rabs_right (/ sum_f_R0 An n))...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1)) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply Rle_lt_trans with
    (sum_f_R0 (fun i:nat => Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
      (n - S N1) * / sum_f_R0 An n)...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1)) * / sum_f_R0 An n <=
sum_f_R0
  (fun i : nat =>
   Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
  (n - S N1) * / sum_f_R0 An n
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat =>
   Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  do 2 rewrite <- (Rmult_comm (/ sum_f_R0 An n)); apply Rmult_le_compat_l...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
0 <= / sum_f_R0 An n
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1)) <=
sum_f_R0
  (fun i : nat =>
   Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
  (n - S N1)
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat =>
   Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  left; apply Rinv_0_lt_compat...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
Rabs
  (sum_f_R0
     (fun i : nat =>
      An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
     (n - S N1)) <=
sum_f_R0
  (fun i : nat =>
   Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
  (n - S N1)
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat =>
   Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply
    (Rsum_abs (fun i:nat => An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))
      (n - S N1))...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat =>
   Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply Rle_lt_trans with
    (sum_f_R0 (fun i:nat => An (S N1 + i)%nat * (eps / 2)) (n - S N1) *
      / sum_f_R0 An n)...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat =>
   Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
  (n - S N1) * / sum_f_R0 An n <=
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1) * / sum_f_R0 An n
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  do 2 rewrite <- (Rmult_comm (/ sum_f_R0 An n)); apply Rmult_le_compat_l...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
0 <= / sum_f_R0 An n
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat =>
   Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
  (n - S N1) <=
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1)
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  left; apply Rinv_0_lt_compat...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat =>
   Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
  (n - S N1) <=
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1)
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply sum_Rle; intros; rewrite Rabs_mult;
    pattern (An (S N1 + n0)%nat) at 2;
      rewrite <- (Rabs_right (An (S N1 + n0)%nat))...An, Bn:nat -> R
l:R
H0:forall n1 : nat, 0 < An n1
H1:cv_infty (fun n1 : nat => sum_f_R0 An n1)
eps:R
H2:eps > 0
H3:forall n1 : nat, 0 < sum_f_R0 An n1
H4:forall n1 : nat, sum_f_R0 An n1 <> 0
H5:Un_cv (fun n1 : nat => / sum_f_R0 An n1) 0
H6:0 < eps / 2
N1:nat
H:forall n1 : nat,
(n1 >= N1)%nat -> Rabs (Bn n1 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n1 : nat,
(N2 <= n1)%nat -> C / sum_f_R0 An n1 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
n0:nat
H10:(n0 <= n - S N1)%nat
Rabs (An (S N1 + n0)%nat) *
Rabs (Bn (S N1 + n0)%nat - l) <=
Rabs (An (S N1 + n0)%nat) * (eps / 2)
An, Bn:nat -> R
l:R
H0:forall n1 : nat, 0 < An n1
H1:cv_infty (fun n1 : nat => sum_f_R0 An n1)
eps:R
H2:eps > 0
H3:forall n1 : nat, 0 < sum_f_R0 An n1
H4:forall n1 : nat, sum_f_R0 An n1 <> 0
H5:Un_cv (fun n1 : nat => / sum_f_R0 An n1) 0
H6:0 < eps / 2
N1:nat
H:forall n1 : nat,
(n1 >= N1)%nat -> Rabs (Bn n1 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n1 : nat,
(N2 <= n1)%nat -> C / sum_f_R0 An n1 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
n0:nat
H10:(n0 <= n - S N1)%nat
An (S N1 + n0)%nat >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply Rmult_le_compat_l...An, Bn:nat -> R
l:R
H0:forall n1 : nat, 0 < An n1
H1:cv_infty (fun n1 : nat => sum_f_R0 An n1)
eps:R
H2:eps > 0
H3:forall n1 : nat, 0 < sum_f_R0 An n1
H4:forall n1 : nat, sum_f_R0 An n1 <> 0
H5:Un_cv (fun n1 : nat => / sum_f_R0 An n1) 0
H6:0 < eps / 2
N1:nat
H:forall n1 : nat,
(n1 >= N1)%nat -> Rabs (Bn n1 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n1 : nat,
(N2 <= n1)%nat -> C / sum_f_R0 An n1 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
n0:nat
H10:(n0 <= n - S N1)%nat
0 <= Rabs (An (S N1 + n0)%nat)
An, Bn:nat -> R
l:R
H0:forall n1 : nat, 0 < An n1
H1:cv_infty (fun n1 : nat => sum_f_R0 An n1)
eps:R
H2:eps > 0
H3:forall n1 : nat, 0 < sum_f_R0 An n1
H4:forall n1 : nat, sum_f_R0 An n1 <> 0
H5:Un_cv (fun n1 : nat => / sum_f_R0 An n1) 0
H6:0 < eps / 2
N1:nat
H:forall n1 : nat,
(n1 >= N1)%nat -> Rabs (Bn n1 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n1 : nat,
(N2 <= n1)%nat -> C / sum_f_R0 An n1 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
n0:nat
H10:(n0 <= n - S N1)%nat
Rabs (Bn (S N1 + n0)%nat - l) <= eps / 2
An, Bn:nat -> R
l:R
H0:forall n1 : nat, 0 < An n1
H1:cv_infty (fun n1 : nat => sum_f_R0 An n1)
eps:R
H2:eps > 0
H3:forall n1 : nat, 0 < sum_f_R0 An n1
H4:forall n1 : nat, sum_f_R0 An n1 <> 0
H5:Un_cv (fun n1 : nat => / sum_f_R0 An n1) 0
H6:0 < eps / 2
N1:nat
H:forall n1 : nat,
(n1 >= N1)%nat -> Rabs (Bn n1 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n1 : nat,
(N2 <= n1)%nat -> C / sum_f_R0 An n1 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
n0:nat
H10:(n0 <= n - S N1)%nat
An (S N1 + n0)%nat >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply Rabs_pos...An, Bn:nat -> R
l:R
H0:forall n1 : nat, 0 < An n1
H1:cv_infty (fun n1 : nat => sum_f_R0 An n1)
eps:R
H2:eps > 0
H3:forall n1 : nat, 0 < sum_f_R0 An n1
H4:forall n1 : nat, sum_f_R0 An n1 <> 0
H5:Un_cv (fun n1 : nat => / sum_f_R0 An n1) 0
H6:0 < eps / 2
N1:nat
H:forall n1 : nat,
(n1 >= N1)%nat -> Rabs (Bn n1 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n1 : nat,
(N2 <= n1)%nat -> C / sum_f_R0 An n1 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
n0:nat
H10:(n0 <= n - S N1)%nat
Rabs (Bn (S N1 + n0)%nat - l) <= eps / 2
An, Bn:nat -> R
l:R
H0:forall n1 : nat, 0 < An n1
H1:cv_infty (fun n1 : nat => sum_f_R0 An n1)
eps:R
H2:eps > 0
H3:forall n1 : nat, 0 < sum_f_R0 An n1
H4:forall n1 : nat, sum_f_R0 An n1 <> 0
H5:Un_cv (fun n1 : nat => / sum_f_R0 An n1) 0
H6:0 < eps / 2
N1:nat
H:forall n1 : nat,
(n1 >= N1)%nat -> Rabs (Bn n1 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n1 : nat,
(N2 <= n1)%nat -> C / sum_f_R0 An n1 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
n0:nat
H10:(n0 <= n - S N1)%nat
An (S N1 + n0)%nat >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  left; apply H; unfold ge; apply le_trans with (S N1);
    [ apply le_n_Sn | apply le_plus_l ]...An, Bn:nat -> R
l:R
H0:forall n1 : nat, 0 < An n1
H1:cv_infty (fun n1 : nat => sum_f_R0 An n1)
eps:R
H2:eps > 0
H3:forall n1 : nat, 0 < sum_f_R0 An n1
H4:forall n1 : nat, sum_f_R0 An n1 <> 0
H5:Un_cv (fun n1 : nat => / sum_f_R0 An n1) 0
H6:0 < eps / 2
N1:nat
H:forall n1 : nat,
(n1 >= N1)%nat -> Rabs (Bn n1 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n1 : nat,
(N2 <= n1)%nat -> C / sum_f_R0 An n1 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
n0:nat
H10:(n0 <= n - S N1)%nat
An (S N1 + n0)%nat >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply Rle_ge; left...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0
  (fun i : nat => An (S N1 + i)%nat * (eps / 2))
  (n - S N1) * / sum_f_R0 An n < eps * / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  rewrite <- (scal_sum (fun i:nat => An (S N1 + i)%nat) (n - S N1) (eps / 2));
    unfold Rdiv; repeat rewrite Rmult_assoc; apply Rmult_lt_compat_l...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ 2 *
(sum_f_R0 (fun i : nat => An (S N1 + i)%nat)
   (n - S N1) * / sum_f_R0 An n) < / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  pattern (/ 2) at 2; rewrite <- Rmult_1_r; apply Rmult_lt_compat_l...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
0 < / 2
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0 (fun i : nat => An (S N1 + i)%nat) (n - S N1) *
/ sum_f_R0 An n < 1
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply Rinv_0_lt_compat; prove_sup...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0 (fun i : nat => An (S N1 + i)%nat) (n - S N1) *
/ sum_f_R0 An n < 1
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  rewrite Rmult_comm; apply Rmult_lt_reg_l with (sum_f_R0 An n)...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0 An n *
(/ sum_f_R0 An n *
 sum_f_R0 (fun i : nat => An (S N1 + i)%nat)
   (n - S N1)) < sum_f_R0 An n * 1
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
1 *
sum_f_R0 (fun i : nat => An (S N1 + i)%nat) (n - S N1) <
sum_f_R0 An n * 1
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  rewrite Rmult_1_l; rewrite Rmult_1_r; rewrite (tech2 An N1 n)...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
sum_f_R0 (fun i : nat => An (S N1 + i)%nat) (n - S N1) <
sum_f_R0 An N1 +
sum_f_R0 (fun i : nat => An (S N1 + i)%nat) (n - S N1)
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  rewrite Rplus_comm;
    pattern (sum_f_R0 (fun i:nat => An (S N1 + i)%nat) (n - S N1)) at 1;
      rewrite <- Rplus_0_r; apply Rplus_lt_compat_l...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (Bn n0 - l) < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
H9:(N1 < n)%nat
/ sum_f_R0 An n >= 0
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  apply Rle_ge; left; apply Rinv_0_lt_compat...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
  replace (sum_f_R0 (fun k:nat => An k * (Bn k - l)) n) with
  (sum_f_R0 (fun k:nat => An k * Bn k) n +
    sum_f_R0 (fun k:nat => An k * - l) n)...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
(sum_f_R0 (fun k : nat => An k * Bn k) n +
 sum_f_R0 (fun k : nat => An k * - l) n) /
sum_f_R0 An n =
sum_f_R0 (fun k : nat => An k * Bn k) n /
sum_f_R0 An n - l
An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * Bn k) n +
sum_f_R0 (fun k : nat => An k * - l) n =
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n
  rewrite <- (scal_sum An n (- l)); field...An, Bn:nat -> R
l:R
H0:forall n0 : nat, 0 < An n0
H1:cv_infty (fun n0 : nat => sum_f_R0 An n0)
eps:R
H2:eps > 0
H3:forall n0 : nat, 0 < sum_f_R0 An n0
H4:forall n0 : nat, sum_f_R0 An n0 <> 0
H5:Un_cv (fun n0 : nat => / sum_f_R0 An n0) 0
H6:0 < eps / 2
N1:nat
H:forall n0 : nat,
(n0 >= N1)%nat -> R_dist (Bn n0) l < eps / 2
C:=Rabs
  (sum_f_R0 (fun k : nat => An k * (Bn k - l)) N1):R
N2:nat
H7:forall n0 : nat,
(N2 <= n0)%nat -> C / sum_f_R0 An n0 < eps / 2
N:=Nat.max N1 N2:nat
n:nat
H8:(n >= S N)%nat
sum_f_R0 (fun k : nat => An k * Bn k) n +
sum_f_R0 (fun k : nat => An k * - l) n =
sum_f_R0 (fun k : nat => An k * (Bn k - l)) n 
  rewrite <- plus_sum; apply sum_eq; intros; ring...
Qed.

Lemma Cesaro_1 :
  forall (An:nat -> R) (l:R),
    Un_cv An l -> Un_cv (fun n:nat => sum_f_R0 An (pred n) / INR n) l.forall (An : nat -> R) (l : R),
Un_cv An l ->
Un_cv
  (fun n : nat =>
   sum_f_R0 An (Init.Nat.pred n) / INR n) l
Proof with trivial.forall (An : nat -> R) (l : R),
Un_cv An l ->
Un_cv
  (fun n : nat =>
   sum_f_R0 An (Init.Nat.pred n) / INR n) l
  intros Bn l H; set (An := fun _:nat => 1)...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  assert (H0 : forall n:nat, 0 < An n)...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
forall n : nat, 0 < An n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  intro; unfold An; apply Rlt_0_1...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  assert (H1 : forall n:nat, 0 < sum_f_R0 An n)...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
forall n : nat, 0 < sum_f_R0 An n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  intro; apply tech1...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  assert (H2 : cv_infty (fun n:nat => sum_f_R0 An n))...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
cv_infty (fun n : nat => sum_f_R0 An n)
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  unfold cv_infty; intro; destruct (Rle_dec M 0) as [Hle|Hnle]...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
M:R
Hle:M <= 0
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < sum_f_R0 An n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
M:R
Hnle:~ M <= 0
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < sum_f_R0 An n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  exists 0%nat; intros; apply Rle_lt_trans with 0...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
M:R
Hnle:~ M <= 0
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < sum_f_R0 An n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  assert (H2 : 0 < M)...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
M:R
Hnle:~ M <= 0
0 < M
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
M:R
Hnle:~ M <= 0
H2:0 < M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < sum_f_R0 An n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  auto with real...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
M:R
Hnle:~ M <= 0
H2:0 < M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < sum_f_R0 An n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  clear Hnle; set (m := up M); elim (archimed M); intros;
    assert (H5 : (0 <= m)%Z)...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
M:R
H2:0 < M
m:=up M:Z
H3:IZR (up M) > M
H4:IZR (up M) - M <= 1
(0 <= m)%Z
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
M:R
H2:0 < M
m:=up M:Z
H3:IZR (up M) > M
H4:IZR (up M) - M <= 1
H5:(0 <= m)%Z
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < sum_f_R0 An n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  apply le_IZR; unfold m; simpl; left; apply Rlt_trans with M...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
M:R
H2:0 < M
m:=up M:Z
H3:IZR (up M) > M
H4:IZR (up M) - M <= 1
H5:(0 <= m)%Z
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < sum_f_R0 An n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  elim (IZN _ H5); intros; exists x; intros; unfold An; rewrite sum_cte;
    rewrite Rmult_1_l; apply Rlt_trans with (IZR (up M))...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
M:R
H2:0 < M
m:=up M:Z
H3:IZR (up M) > M
H4:IZR (up M) - M <= 1
H5:(0 <= m)%Z
x:nat
H6:m = Z.of_nat x
n:nat
H7:(x <= n)%nat
IZR (up M) < INR (S n)
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  apply Rle_lt_trans with (INR x)...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
M:R
H2:0 < M
m:=up M:Z
H3:IZR (up M) > M
H4:IZR (up M) - M <= 1
H5:(0 <= m)%Z
x:nat
H6:m = Z.of_nat x
n:nat
H7:(x <= n)%nat
IZR (up M) <= INR x
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
M:R
H2:0 < M
m:=up M:Z
H3:IZR (up M) > M
H4:IZR (up M) - M <= 1
H5:(0 <= m)%Z
x:nat
H6:m = Z.of_nat x
n:nat
H7:(x <= n)%nat
INR x < INR (S n)
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  rewrite INR_IZR_INZ; fold m; rewrite <- H6; right...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
M:R
H2:0 < M
m:=up M:Z
H3:IZR (up M) > M
H4:IZR (up M) - M <= 1
H5:(0 <= m)%Z
x:nat
H6:m = Z.of_nat x
n:nat
H7:(x <= n)%nat
INR x < INR (S n)
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  apply lt_INR; apply le_lt_n_Sm...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  assert (H3 := Cesaro _ _ _ H H0 H2)...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n : nat, 0 < An n
H1:forall n : nat, 0 < sum_f_R0 An n
H2:cv_infty (fun n : nat => sum_f_R0 An n)
H3:Un_cv
  (fun n : nat =>
   sum_f_R0 (fun k : nat => An k * Bn k) n /
   sum_f_R0 An n) l
Un_cv
  (fun n : nat =>
   sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
  unfold Un_cv; unfold Un_cv in H3; intros; elim (H3 _ H4); intros;
    exists (S x); intros; unfold R_dist; unfold R_dist in H5;
      apply Rle_lt_trans with
        (Rabs
          (sum_f_R0 (fun k:nat => An k * Bn k) (pred n) / sum_f_R0 An (pred n) - l))...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
Rabs (sum_f_R0 Bn (Init.Nat.pred n) / INR n - l) <=
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k)
     (Init.Nat.pred n) / 
   sum_f_R0 An (Init.Nat.pred n) - l)
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k)
     (Init.Nat.pred n) / 
   sum_f_R0 An (Init.Nat.pred n) - l) < eps
  right;
    replace (sum_f_R0 Bn (pred n) / INR n - l) with
    (sum_f_R0 (fun k:nat => An k * Bn k) (pred n) / sum_f_R0 An (pred n) - l)...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
sum_f_R0 (fun k : nat => An k * Bn k)
  (Init.Nat.pred n) / sum_f_R0 An (Init.Nat.pred n) -
l = sum_f_R0 Bn (Init.Nat.pred n) / INR n - l
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k)
     (Init.Nat.pred n) / 
   sum_f_R0 An (Init.Nat.pred n) - l) < eps
  unfold Rminus; do 2 rewrite <- (Rplus_comm (- l));
    apply Rplus_eq_compat_l...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
sum_f_R0 (fun k : nat => An k * Bn k)
  (Init.Nat.pred n) / sum_f_R0 An (Init.Nat.pred n) =
sum_f_R0 Bn (Init.Nat.pred n) / INR n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k)
     (Init.Nat.pred n) / 
   sum_f_R0 An (Init.Nat.pred n) - l) < eps
  unfold An;
    replace (sum_f_R0 (fun k:nat => 1 * Bn k) (pred n)) with
    (sum_f_R0 Bn (pred n))...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
sum_f_R0 Bn (Init.Nat.pred n) /
sum_f_R0 (fun _ : nat => 1) (Init.Nat.pred n) =
sum_f_R0 Bn (Init.Nat.pred n) / INR n
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
sum_f_R0 Bn (Init.Nat.pred n) =
sum_f_R0 (fun k : nat => 1 * Bn k) (Init.Nat.pred n)
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k)
     (Init.Nat.pred n) / 
   sum_f_R0 An (Init.Nat.pred n) - l) < eps
  rewrite sum_cte; rewrite Rmult_1_l; replace (S (pred n)) with n...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
n = S (Init.Nat.pred n)
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
sum_f_R0 Bn (Init.Nat.pred n) =
sum_f_R0 (fun k : nat => 1 * Bn k) (Init.Nat.pred n)
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k)
     (Init.Nat.pred n) / 
   sum_f_R0 An (Init.Nat.pred n) - l) < eps
  apply S_pred with 0%nat; apply lt_le_trans with (S x)...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
(0 < S x)%nat
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
sum_f_R0 Bn (Init.Nat.pred n) =
sum_f_R0 (fun k : nat => 1 * Bn k) (Init.Nat.pred n)
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k)
     (Init.Nat.pred n) / 
   sum_f_R0 An (Init.Nat.pred n) - l) < eps
  apply lt_O_Sn...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
sum_f_R0 Bn (Init.Nat.pred n) =
sum_f_R0 (fun k : nat => 1 * Bn k) (Init.Nat.pred n)
Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k)
     (Init.Nat.pred n) / 
   sum_f_R0 An (Init.Nat.pred n) - l) < eps
  apply sum_eq; intros; ring...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k)
     (Init.Nat.pred n) / 
   sum_f_R0 An (Init.Nat.pred n) - l) < eps
  apply H5; unfold ge; apply le_S_n; replace (S (pred n)) with n...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
n = S (Init.Nat.pred n)
  apply S_pred with 0%nat; apply lt_le_trans with (S x)...Bn:nat -> R
l:R
H:Un_cv Bn l
An:=fun _ : nat => 1:nat -> R
H0:forall n0 : nat, 0 < An n0
H1:forall n0 : nat, 0 < sum_f_R0 An n0
H2:cv_infty (fun n0 : nat => sum_f_R0 An n0)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
     sum_f_R0 An n0) l < eps0
eps:R
H4:eps > 0
x:nat
H5:forall n0 : nat,
(n0 >= x)%nat ->
Rabs
  (sum_f_R0 (fun k : nat => An k * Bn k) n0 /
   sum_f_R0 An n0 - l) < eps
n:nat
H6:(n >= S x)%nat
(0 < S x)%nat
  apply lt_O_Sn...
Qed.