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

(**********)
Lemma sum_eq_R0 :
forall (An:nat -> R) (N:nat),
(forall n:nat, (n <= N)%nat -> An n = 0) -> sum_f_R0 An N = 0.