proposition: Lemma nlim_sum {I : eqType} (u : I -> nat -> R) (r : seq I) : (forall i, i \in r -> iscvg (u i)) -> nlim (fun n => \sum_(i <- r) (u i) n) = (\sum_(i <- r) nlim (u i))%E.
proof: Proof. elim: r => /= [|i r ih] h; first rewrite big_nil. by rewrite (@eq_nlim 0%:S) ?nlimC //= => n; rewrite big_nil. rewrite big_cons -ih -?nlimD. by move=> j jr; apply/h; rewrite inE jr orbT. by apply/h; rewrite inE eqxx. by apply/iscvg_sum=> j jr; apply/h; rewrite inE jr orbT. by apply/eq_nlim=> n /=; rewrite big_cons. Qed.
imports: mathcomp(all_ssreflect all_algebra) mathcomp.bigenough.bigenough mathcomp.classical(boolp classical_sets functions) mathcomp.classical(mathcomp_extra) xfinmap ereal reals discrete topology