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.