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 maj_by_pos : forall Un:nat -> R, { l:R | Un_cv Un l } -> exists l : R, 0 < l /\ (forall n:nat, Rabs (Un n) <= l).