In Luenberger book Cauchy-Schwarz Inequality is defined like this: For all $x,y$ in an inner product space $|(x|y)| \le \|x\|\|y\|$. Equality holds if and only if $x = \lambda y$ or $y = \theta$.
Proof starts for all scalars $\lambda$,
$ 0 \le (x-\lambda y | x-\lambda y) = (x|x) - \lambda(y|x) - \bar{\lambda}(x|y) + |\lambda|^2 (y|y) $
I understand this expansion. But then, it will select a particular $\lambda = (x|y)/(y|y)$, and obtains
$ 0 \le (x|x) - \frac{|(x|y)|^2}{(y|y)} $
I dont understand how he chose that particular $\lambda$. I guess I understand why, he chose it to get rid of it in the main equation, but is it okay to chose any $\lambda$ that will clean up the equation like this?