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 eventually_subseq_loc : forall phi, eventually (fun n => (phi n < phi (S n))%nat) -> filterlim phi eventually eventually.