(*****************************************************************)
(**           Convergence properties of sequences                *)
(*****************************************************************)

Definition Un_decreasing (Un:nat -> R) : Prop :=
  forall n:nat, Un (S n) <= Un n.Definition opp_seq (Un:nat -> R) (n:nat) : R := - Un n.Definition has_ub (Un:nat -> R) : Prop := bound (EUn Un).Definition has_lb (Un:nat -> R) : Prop := bound (EUn (opp_seq Un)).

Lemma ub_to_lub : forall Un:nat -> R, has_ub Un -> { l:R | is_lub (EUn Un) l }.