(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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)         *)
(************************************************************************)
Local Open Scope R_scope.
Implicit Type r : R.
(* classical is needed for [Un_cv_crit] *)
(*********************************************************)
(** *        Definition of sequence and properties       *)
(*                                                       *)
(*********************************************************)
Section sequence.
(*********)
Variable Un : nat -> R.
(*********)
Fixpoint Rmax_N (N:nat) : R :=
match N with
| O => Un 0
| S n => Rmax (Un (S n)) (Rmax_N n)
end.
(*********)
Definition EUn r : Prop :=  exists i : nat, r = Un i.
(*********)
Definition Un_cv (l:R) : Prop :=
forall eps:R,
eps > 0 ->
exists N : nat, (forall n:nat, (n >= N)%nat -> Rdist (Un n) l < eps).
(*********)
Definition Cauchy_crit : Prop :=
forall eps:R,
eps > 0 ->
exists N : nat,
(forall n m:nat,
(n >= N)%nat -> (m >= N)%nat -> Rdist (Un n) (Un m) < eps).
(*********)
Definition Un_growing : Prop := forall n:nat, Un n <= Un (S n).

(*********)
Lemma growing_prop :
forall n m:nat, Un_growing -> (n >= m)%nat -> Un n >= Un m.