*Problem:*

Let be positive real numbers. Prove the Inequality

*Solution:*

First, we observe that from the AM-GM Inequality we have that

or equivalently

and now taking square roots for both sides, we see that

Therefore, we acquire

Now, from the Cauchy-Schwarz Inequality we have for the right hand side

Moreover, once again, from the AM-GM Inequality we have for the denominators of the above sum that

and thus, it will hold that

Finally, we have proved that

The Inequality we have proved is

which also is the Inequality we are given to prove. The proof is completedÂ *Q.E.D.*

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