We now prove that <math>-\infty . Notice by assumption there exists at least one formula_20 for which formula_33, and so <math>-\infty . However, it may be the case that there are no formula_20 for which formula_26, in which case formula_36 and the inequality is trivial (in this case notice that the third case above cannot happen). Therefore, we may assume that formula_37 and there is at least one formula_20 for which formula_26. To prove the inequality, it suffices to show that whenever formula_20 and formula_33, and formula_42 and formula_43, then formula_44. Indeed,