(** Subsequences *)

(* Specifications: *)
Lemma eventually_subseq_loc :
forall phi,
eventually (fun n => (phi n < phi (S n))%nat) ->
filterlim phi eventually eventually.
Lemma eventually_subseq :
forall phi,
(forall n, (phi n < phi (S n))%nat) ->
filterlim phi eventually eventually.
Lemma is_lim_seq_subseq (u : nat -> R) (l : Rbar) (phi : nat -> nat) :
filterlim phi eventually eventually ->
is_lim_seq u l ->
is_lim_seq (fun n => u (phi n)) l.
Lemma ex_lim_seq_subseq (u : nat -> R) (phi : nat -> nat) :
filterlim phi eventually eventually ->
ex_lim_seq u ->
ex_lim_seq (fun n => u (phi n)).
Lemma Lim_seq_subseq (u : nat -> R) (phi : nat -> nat) :
filterlim phi eventually eventually ->
ex_lim_seq u ->
Lim_seq (fun n => u (phi n)) = Lim_seq u.
Lemma is_lim_seq_incr_1 (u : nat -> R) (l : Rbar) :
is_lim_seq u l <-> is_lim_seq (fun n => u (S n)) l.
Lemma ex_lim_seq_incr_1 (u : nat -> R) :
ex_lim_seq u <-> ex_lim_seq (fun n => u (S n)).
Lemma Lim_seq_incr_1 (u : nat -> R) :
Lim_seq (fun n => u (S n)) = Lim_seq u.
Lemma is_lim_seq_incr_n (u : nat -> R) (N : nat) (l : Rbar) :
is_lim_seq u l <-> is_lim_seq (fun n => u (n + N)%nat) l.
Lemma ex_lim_seq_incr_n (u : nat -> R) (N : nat) :
ex_lim_seq u <-> ex_lim_seq (fun n => u (n + N)%nat).
Lemma Lim_seq_incr_n (u : nat -> R) (N : nat) :
Lim_seq (fun n => u (n + N)%nat) = Lim_seq u.