(** * 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_Finite (E : Rbar -> Prop) (l : R) : Rbar_is_lower_bound E (Finite l) -> is_upper_bound (fun x => E (Finite (- x))) (- l).