proposition: Lemma lee_nneseries (R : realType) (u v : (\bar R)^nat) (P : pred nat) N : (forall i, P i -> 0 <= u i) -> (forall n, P n -> u n <= v n) -> \sum_(N <= i <oo | P i) u i <= \sum_(N <= i <oo | P i) v i.
proof: Proof. move=> u0 Puv; apply: lee_lim. - by apply: is_cvg_ereal_nneg_natsum_cond => n ? /u0; exact. - apply: is_cvg_ereal_nneg_natsum_cond => n _ Pn. by rewrite (le_trans _ (Puv _ Pn))// u0. - by near=> n; apply: lee_sum => k; exact: Puv. Unshelve. all: by end_near. Qed.
imports: mathcomp(all_ssreflect ssralg ssrint ssrnum matrix) mathcomp(interval rat) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(set_interval) reals ereal signed topology normedtype landau