unfold rat_inf_limit, inf_limit in *.
intuition.
destruct (H epsilon); intuition.
destruct (H0 epsilon); intuition.
exists (max c (max x x0)).
intuition.
destruct (H3 n').
destruct (H4 n').
eapply leRat_trans.
eapply ratDistance_le_max.
eapply H1.
eapply Max.max_lub_l.
eauto.
eapply H9.
eapply H10.
eapply H2.
eapply Max.max_lub_l.
eauto.
eapply H9.
eapply H11.
eapply maxRat_leRat_same; eauto using Max.max_lub_l, Max.max_lub_r.
Qed.

Lemma rat_inf_limit_div_2 : forall (f : nat -> Rat -> Prop)(v : Rat),
rat_inf_limit f v ->
rat_inf_limit (fun n => (f (div2 n))) v.