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