If are positive real numbers such that , then prove that
From the Cauchy-Schwarz inequality we deduce that
Removing the square we get that
. Let us now divide by .
Then we have .
Moreover, multiply by . We, thus, acquire
So,we have proved that
We will now apply Holder’s Inequality, that is
, or .
Rewrite the sum as .
Then we get that:
And finally, from the AM-GM inequality we have
So, we have proved that