Definition ex_lim_seq_cauchy (u : nat -> R) :=
  forall eps : posreal, exists N : nat, forall n m,
    (N <= n)%nat -> (N <= m)%nat -> Rabs (u n - u m) < eps.

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.