I'm currently going through Spivak's Calculus(3rd edition) and I'm almost certain that I have found a mistake in one of the exercises. In Chapter 1 Exercise 19 (a), the Schwarz Inequality is given by:
$x_1 y_1 + x_2 y_2 \leq \sqrt{x_1^2 + x_2^2} \sqrt{y_1^2 + y_2^2}$
Prove that if $x_1 = \lambda y_1$ and $x_2 = \lambda y_2$ for some number $\lambda$ then equality holds in the Schwarz inequality.
My proof so far:
$x_1 y_1 + x_2 y_2 = \lambda y_1 y_1 + \lambda y_2 y_2 = \lambda y_1^2 + \lambda y_2^2 = \lambda (y_1^2 + y_2^2)$.
So here's where the trouble starts for me: $(y_1^2 + y_2^2) \geq 0$, but we didn't make any assumptions for $\lambda$. In order for $\lambda (y_1^2 + y_2^2) = \sqrt{\lambda^2 (y_1^2 + y_2^2)^2}$,
$\lambda \geq0$ must hold. I think that the problem statement is wrong and that $\lambda$ must be positive.Is this really the case or am I wrong?