Section sequence.
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Variable Un : nat -> R.
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Fixpoint Rmax_N (N:nat) : R :=
match N with
| O => Un 0
| S n => Rmax (Un (S n)) (Rmax_N n)
end.
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Definition EUn r : Prop :=  exists i : nat, r = Un i.
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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).
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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).
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Definition Un_growing : Prop := forall n:nat, Un n <= Un (S n).

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Lemma Un_cv_crit_lub : Un_growing -> forall l, is_lub EUn l -> Un_cv l.