(** ** Least upper bound and Greatest Lower Bound *)

Definition Rbar_is_lub (E : Rbar -> Prop) (l : Rbar) :=
  Rbar_is_upper_bound E l /\
    (forall b : Rbar, Rbar_is_upper_bound E b -> Rbar_le l b).Definition Rbar_is_glb (E : Rbar -> Prop) (l : Rbar) :=
  Rbar_is_lower_bound E l /\
    (forall b : Rbar, Rbar_is_lower_bound E b -> Rbar_le b l).

Lemma Rbar_glb_lub (E : Rbar -> Prop) (l : Rbar) : Rbar_is_glb (fun x => E (Rbar_opp x)) (Rbar_opp l) <-> Rbar_is_lub E l.