proposition: Lemma nneseriesD (R : realType) (f g : nat -> \bar R) (P : pred nat) : (forall i, P i -> 0 <= f i) -> (forall i, P i -> 0 <= g i) -> \sum_(i <oo | P i) (f i + g i) = \sum_(i <oo | P i) f i + \sum_(i <oo | P i) g i.
proof: Proof. move=> f_eq0 g_eq0. transitivity (limn (fun n => \sum_(0 <= i < n | P i) f i + \sum_(0 <= i < n | P i) g i)). by apply/congr_lim/funext => n; rewrite big_split. rewrite limeD /adde_def //=; do ? exact: is_cvg_nneseries. by rewrite ![_ == -oo]gt_eqF ?andbF// (@lt_le_trans _ _ 0) ?[_ < _]real0// nneseries_ge0. 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