(** * Bounds for sets in [Rbar] *)
(** ** Upper and Lower bounds *)

Definition Rbar_is_upper_bound (E : Rbar -> Prop) (l : Rbar) :=
  forall x, E x -> Rbar_le x l.Definition Rbar_is_lower_bound (E : Rbar -> Prop) (l : Rbar) :=
  forall x, E x -> Rbar_le l x.

Lemma Rbar_lb_ub (E : Rbar -> Prop) (l : Rbar) : Rbar_is_lower_bound (fun x => E (Rbar_opp x)) (Rbar_opp l) <-> Rbar_is_upper_bound E l.