(** ** 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.