absurd (N >= 2).
elim s; intro y; rewrite y; unfold ge in |- *; auto with arith real.
exact H.
case (exist_last N g); intros; elim a; intros; auto with arith real.
Qed.

(**********)
Lemma first_n :
forall N n : nat,
N >= 2 ->
0 < n ->
n < N ->
(0 < alpha)%R /\ (alpha < 1)%R ->
(frac_part_n_alpha (first N) <= frac_part_n_alpha n)%R.