In the problem section, there is a problem asking the reader to prove
$$|(f,g)|\le \|f\|\cdot\|g\|$$
where $|(f,g)|$ is the inner product and $\|f\|$ is the norm by defining the function $F(\alpha) = \|f+\alpha g\|^2$ and (I assume this is how they intended) showing how there is at most one root for $\alpha$ and so the discriminant of $F \le 0$ where then the proof follows quickly. It does seem like a simple proof, but I'm wondering if it would be more intuitive to define the inner product (and the norm) as a discrete Riemann sum, expanding the product on the RHS of the Schwarz inequality, then cancelling, working backwards, and taking the limit of the Riemann sum. Is this flawed in some way or is this an actual approach?