unfold rat_inf_limit, inf_limit.
intuition.
destruct (H epsilon); intuition.
econstructor.
intuition.
specialize (H1 (div2 n')).
eapply H1.
eapply div2_ge; eauto.
eauto.
Qed.

Lemma rat_inf_limit_sum : forall (f1 f2 : nat -> Rat -> Prop) v1 v2,
left_total f1 ->
left_total f2 ->
rat_inf_limit f1 v1 ->
rat_inf_limit f2 v2 ->
rat_inf_limit (fun n => (ratAdd_rel (f1 n) (f2 n))) (v1 + v2).