(** Functions *)

Definition LimSup_seq (u : nat -> R) :=
  proj1_sig (ex_LimSup_seq u).Definition LimInf_seq (u : nat -> R) :=
  proj1_sig (ex_LimInf_seq u).

Lemma LimSup_InfSup_seq (u : nat -> R) : LimSup_seq u = Inf_seq (fun m => Sup_seq (fun n => u (n + m)%nat)).