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**RogueDemon** 19. (a) Prove that if $\displaystyle x_1 = ny_1$ and $\displaystyle x_2 = ny_2$ for some number $\displaystyle n$, then equality holds in the Schwarz inequality. Prove the same thing if $\displaystyle y_1 = y_2 = 0$. Now suppose that $\displaystyle y_1$ and $\displaystyle y_2$ are not both $\displaystyle 0$, and that there is no number $\displaystyle n$ such that $\displaystyle x_1 = ny_1$ and $\displaystyle x_2 = ny_2$. Then

$\displaystyle 0 < (ny_1 - x_1)^2 + (ny_2 - x_2)^2 $

$\displaystyle = n^2(y_1^2 + y_2^2) - 2n(x_1y_1 + x_2y_2) + (x_1^2 + y_2^2).$

Using Problem 18, complete the proof of the Schwarz inequality (i.e. prove that $\displaystyle x_1y_1 + x_2y_2 <= \sqrt{x_1^2 + x_2^2}\sqrt{y_1^2 + y_2^2}$).

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 $\displaystyle x_1y_1 + x_2y_2 < \frac{1}{2n}(x_1^2 + y_2^2) + \frac{n}{2}(y_1^2 + y_2^2)$, but that doesn't seem to work either.