Let be non-negative numbers, no two of them are zero. Prove that
Making the Cauchy reverse technique we have that . Doing that cyclic over the fractions we need to prove that
Let us now multiply each fraction by respectively and apply Cauchy-Schwartz inequality. We get that:
, or .
So, the current inequality reduces to
Doing some manipulations we have that
Now we have to expand the sums. So:
Back to our inequality now, if we substitute those sums we reach to the obvious conclusion