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