(** * Limit of sequences *)
(** ** Definition *)
Definition is_lim_seq (u : nat -> R) (l : Rbar) :=
filterlim u eventually (Rbar_locally l).
Definition is_lim_seq' (u : nat -> R) (l : Rbar) :=
match l with
| Finite l => forall eps : posreal, eventually (fun n => Rabs (u n - l) < eps)
| p_infty => forall M : R, eventually (fun n => M < u n)
| m_infty => forall M : R, eventually (fun n => u n < M)
end.
Definition ex_lim_seq (u : nat -> R) :=
exists l, is_lim_seq u l.
Definition ex_finite_lim_seq (u : nat -> R) :=
exists l : R, is_lim_seq u l.
Definition Lim_seq (u : nat -> R) : Rbar :=
Rbar_div_pos (Rbar_plus (LimSup_seq u) (LimInf_seq u))
{| pos := 2; cond_pos := Rlt_R0_R2 |}.

Lemma is_lim_seq_spec :
forall u l,
is_lim_seq' u l <-> is_lim_seq u l.