I was looking at a number of different proofs of the cauchy schwarz inequality in an inner product space ($\mathbb{R}^n$ or $\mathbb{C}^n$).
All of them used the idea of $||x-sy||$ where $s$ was selected in particular fashion which in the real case, s would be the value that minimized the function $f(s) = ||x-sy||$
The thing I am confused on is that the books say "s was selected so that $||x-sy||$ would be minimized". I don't understand how beforehand minimizing $||x-sy||$ would be known to be relevant to the inequality.
What is the inuitive link between the minimum of $f(s) = ||x-sy||$ and the Cauchy Schwarz inequality?