We know that

.

Let us conjecture that

.

This means that

, that is,

. Therefore, if our conjecture is true, we must have

.

Let us consider the function

. Find its minimum on

. You will find that the minimum of

is

, and this minimum is reached when

.

But we know that

, thus the minimum

is never reached with

.

. You can follow the steps backwards to finally prove that

________________________

That was the nearly-complete proof. Now, you only have to show that the minimum of

is

when

, and you are done