Definition sequence_ub (Un:nat -> R) (pr:has_ub Un)
(i:nat) : R := lub (fun k:nat => Un (i + k)%nat) (maj_ss Un i pr).
Definition sequence_lb (Un:nat -> R) (pr:has_lb Un)
(i:nat) : R := glb (fun k:nat => Un (i + k)%nat) (min_ss Un i pr).
(* Compatibility *)
Notation sequence_majorant := sequence_ub (only parsing).
Notation sequence_minorant := sequence_lb (only parsing).

Lemma tech13 :
forall (An:nat -> R) (k:R),
0 <= k < 1 ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) k ->
exists k0 : R,
k < k0 < 1 /\
(exists N : nat,
(forall n:nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)).