For all non-negative real numbers with sum , prove that
Assume without loss of generality that
. Moreover, denote by the .
Then we get that
. From the hypothesis we deduce also that .
Let us now transform the factors of the inequality in terms of .
Thus we have that:
Define by the Left Hand Side of the Inequality, that is
We will now prove that . Denote the Left Hand Side by . We must prove that .
We know that
We claim that the second factor of is greater than zero. Indeed, this is true as since we get that:
Multiplying these Inequalities we have that or . Adding up the we get that
Thus we have prove that
But from the last inequality we deduce that
due to the maximized value of .
This completes the first scale of the proof. Now we only need to prove that if
which is obviously true since it is of the form , Q.E.D.