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

Definition is_lub_Rbar (E : R -> Prop) (l : Rbar) :=
  is_ub_Rbar E l /\ (forall b, is_ub_Rbar E b -> Rbar_le l b).Definition is_glb_Rbar (E : R -> Prop) (l : Rbar) :=
  is_lb_Rbar E l /\ (forall b, is_lb_Rbar E b -> Rbar_le b l).

Lemma ex_lub_Rbar (E : R -> Prop) : {l : Rbar | is_lub_Rbar E l}.