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

(* Specifications: *)
(* 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.
(**********)
Lemma cv_cauchy_1 :
forall An:nat -> R,
{ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l } ->
Cauchy_crit_series An.
Lemma cv_cauchy_2 :
forall An:nat -> R,
Cauchy_crit_series An ->
{ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
(**********)
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.