Problem:
If are positive real numbers, such that , prove that
.
Solution:
From the AM-GM we deduce that
.
So,
.
Divide the last relation by . Then we get that
,
or
.
So, it is enough to prove that
.
But it holds that,
.
From this we need to prove now
.
That is,
,
which is obvious from the AM-GM inequality, Q.E.D.