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