Let be non-negative numbers. If , prove that
If we square both sides we get
Now from Cauchy-Schwartz inequality we know that .
So, it suffices to prove that
For the inequality is true. So, we only need to prove it for .
Rewrite the inequality in the form
We know that so, it is enough to prove it for .
For we have that
which reduces to
, which is a well known inequality and we have proved it here: Well-known inequality, Q.E.D.