(************************************************************************)
(*         *   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).

(* If (|An|) satisfies the Cauchy's criterion for series, then (An) too *)
Lemma cauchy_abs :
forall An:nat -> R,
Cauchy_crit_series (fun i:nat => Rabs (An i)) -> Cauchy_crit_series An.