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).

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Lemma cauchy_maj : forall Un:nat -> R, Cauchy_crit Un -> has_ub Un.