19. (a) Prove that if

and

for some number

, then equality holds in the Schwarz inequality. Prove the same thing if

. Now suppose that

and

are not both

, and that there is no number

such that

and

. Then

Using Problem 18, complete the proof of the Schwarz inequality (i.e. prove that

).

I've proven the first two assumptions so far (by substitution), but I'm not sure how to prove the third. I know that I have to complete the square (which I've done in problem 18), but I can't seem to do that in this problem since there are double the variables. I've tried manipulating the inequality such that

, but that doesn't seem to work either.