(** * 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_is_lb_subset (E1 E2 : Rbar -> Prop) (l : Rbar) : (forall x, E1 x -> E2 x) -> (Rbar_is_lower_bound E2 l) -> (Rbar_is_lower_bound E1 l).