*Problem:*

Let be positive real numbers such that , and also let be a natural number. Prove that

.

*Solution:*

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

or

.

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.*

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