Definition family_closed_set (f:family) : Prop :=
  forall x:R, closed_set (f x).Definition intersection_vide_in (D:R -> Prop) (f:family) : Prop :=
  forall x:R,
    (ind f x -> included (f x) D) /\
    ~ (exists y : R, intersection_family f y).Definition intersection_vide_finite_in (D:R -> Prop)
  (f:family) : Prop := intersection_vide_in D f /\ family_finite f.

Theorem Bolzano_Weierstrass : forall (un:nat -> R) (X:R -> Prop), compact X -> (forall n:nat, X (un n)) -> exists l : R, ValAdh un l.