unfold continuous_at, rat_limit, rat_inf_limit, inf_limit in *.
intuition.
edestruct H1; eauto.
intuition.
destruct (H x).
trivial.
exists x0.
intuition.
destruct (H2 n').
eapply H6.
eauto.
eapply H4.
eapply H7.
trivial.
Qed.

Lemma rat_inf_limit_trans : forall (f1 f2 : nat -> Rat -> Prop) a,
rat_inf_limit_2 f1 f2 ->
rat_inf_limit f2 a ->
left_total f2 ->
rat_inf_limit f1 a.