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

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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.