induction k; intuition; simpl in *.
symmetry.
eapply num_dem_same_rat1.
trivial.
rewrite IHk.
rewrite <- ratMult_num_den.
eapply eqRat_terms.
intuition.
eapply posnatMult_eq.
Grab Existential Variables.
destruct p.
econstructor.
eapply mult_gt_zero_if; eauto.
Qed.

Lemma expRat_le_half_exists : forall r,
~ 1 <= r ->
exists n, expRat r n <= (1/2).