*Problem:*

Let with . Show that

.

*Solution:*

Subtract each side with . Then, from hypothesis we have that

.

Or

.

From the last relation we have that

.

Assume without loss of generality that .

Then we have that

.

Doing some manipulations on both sides we acquire that

.

Or,

,

which holds.

Indeed, from the AM-GM inequality we have that

and from Cauchy-Schwartz inequality

.

Adding up these inequalities we get the desired result,* Q.E.D.*

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