Lemma sum_maj1 : forall (fn:nat -> R -> R) (An:nat -> R) (x l1 l2:R) (N:nat), 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 - SP fn N x) <= l2 - sum_f_R0 An N.