Prove that with equality if and only if .

* Proof. * First, by Theorem I.20 we know that for any

we have if

, then

. Further, since

, we have that if

, then

. Thus,

and

and so

.

Now, if

, then

. Conversely, if

, then if

we must have

(otherwise the sum would be greater than 0). However, we know

, so this is a contradiction. Therefore,

, and hence,

, as well. By Theorem I.20 again, this must mean

(since if

or

then

or

is greater than 0).

Hence, we indeed have

with equality if and only if

*Related*