proposition: Lemma limn_esup_lim u : limn_esup u = limn (esups u).
proof: Proof. apply/eqP; rewrite eq_le; apply/andP; split. apply: lime_ge; first exact: is_cvg_esups. near=> m; apply: ereal_inf_lb => /=. by exists [set k | (m <= k)%N] => //=; exists m. apply: lb_ereal_inf => /= _ [A [r /= r0 rA] <-]. apply: lime_le; first exact: is_cvg_esups. near=> m; apply: le_ereal_sup => _ [n /= mn] <-. exists n => //; apply: rA => //=; apply: leq_trans mn. by near: m; exists r. 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