Implicit Type r : R.(*********)
  Fixpoint Rmax_N (N:nat) : R :=
    match N with
      | O => Un 0
      | S n => Rmax (Un (S n)) (Rmax_N n)
    end.(*********)
  Definition EUn r : Prop :=  exists i : nat, r = Un i.(*********)
  Definition Un_cv (l:R) : Prop :=
    forall eps:R,
      eps > 0 ->
      exists N : nat, (forall n:nat, (n >= N)%nat -> Rdist (Un n) l < eps).(*********)
  Definition Cauchy_crit : Prop :=
    forall eps:R,
      eps > 0 ->
      exists N : nat,
        (forall n m:nat,
          (n >= N)%nat -> (m >= N)%nat -> Rdist (Un n) (Un m) < eps).(*********)
  Definition Un_growing : Prop := forall n:nat, Un n <= Un (S n).

Lemma Un_bound_imp : forall x:R, (forall n:nat, Un n <= x) -> is_upper_bound EUn x.