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_cv : forall (Un:nat -> R) (pr:Cauchy_crit Un), { l:R | Un_cv (sequence_ub Un (cauchy_maj Un pr)) l }.