(** Notations *)

Definition Sup_seq (u : nat -> Rbar) := proj1_sig (ex_sup_seq u).Definition Inf_seq (u : nat -> Rbar) := proj1_sig (ex_inf_seq u).

Lemma Rbar_sup_eq_lub (u : nat -> Rbar) : Sup_seq u = Rbar_lub (fun x => exists n, x = u n).