intuition.
rattac.
rewrite mult_plus_distr_r.
f_equal.
eapply mult_assoc.
Qed.

Theorem distance_le_prod_f :
forall (f : nat -> Rat) k,
(forall i, | (f i) - (f (S i)) | <= k) ->
forall q0,
| (f 0%nat) - (f q0) | <= q0/1 * k.