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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)         *)
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Local Open Scope R_scope.
(* Cauchy's criterion for series *)
Definition Cauchy_crit_series (An:nat -> R) : Prop :=
Cauchy_crit (fun N:nat => sum_f_R0 An N).
Definition SP (fn:nat -> R -> R) (N:nat) (x:R) : R :=
sum_f_R0 (fun k:nat => fn k x) N.

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Lemma sum_incr :
forall (An:nat -> R) (N:nat) (l:R),
Un_cv (fun n:nat => sum_f_R0 An n) l ->
(forall n:nat, 0 <= An n) -> sum_f_R0 An N <= l.