Let be positive real numbers such that , and also let be a natural number. Prove that
Observe that . Thus, from the AM – GM Inequality we get that
From the above Inequality, we are now capable of building our problem. Specifically, we have that
We must see now that
From the above result we can apply once again the AM – GM Inequality and acquire
Summing up the Inequalities together we get that
Moreover, we have that
which reduces to the , Q.E.D.