Section sequence.
(*********)
Variable Un : nat -> 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 -> R_dist (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 -> R_dist (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.