Prove that the inequality
if the real numbers satisfy , then
The lemma is self-explanatory if we think of as the "saltiness" of a glass of salty water with total mass , and is the mass of the salt. By increasing the numerator and denominator by the same amout (adding salt of mass in the glass) the "saltiness" increases.
By Lemma 1,
since is an increasing function
and this completes our proof.