Let be non-negative numbers, no two of them are zero. Prove that
Let . We have
from which the desired inequality follows. Equality occurs if and only if .
Divide each fraction with respectively. Then we get that
Let us denote . Then the inequality transforms to
Let us now use the transformation which makes the inequality to
From Cauchy’s inequality now, we get that
So, we only need to prove that
which is obviously true. Equality holds only for , Q.E.D.