Definition SP (fn:nat -> R -> R) (N:nat) (x:R) : R :=
  sum_f_R0 (fun k:nat => fn k x) N.

(* Specifications: *)
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.
Lemma sum_cv_maj : forall (An:nat -> R) (fn:nat -> R -> R) (x l1 l2:R), Un_cv (fun n:nat => SP fn n x) l1 -> Un_cv (fun n:nat => sum_f_R0 An n) l2 -> (forall n:nat, Rabs (fn n x) <= An n) -> Rabs l1 <= l2.