Cauchy-Schwarz inequality holds for all symetric semi-definite bilinear forms, so for inner products in particular. The proof is general.
Suppose $(\cdot,\cdot)$ is an inner product on a vector space $E$.
Let $x,y \in E$ and consider the polynomial
P(\lambda) = \|x+\lambda y\|^2 = \|x\|^2 + 2 (x,y) \lambda + \|y\|^2 \lambda^2.
Suppose $y \neq 0$ (otherwise the inequality is immediate).
This polynomial is of degree 2 and is non-negative on all $\mathbb R$ so its discriminant is non-positive, that is:
(2 (x,y))^2 - 4\|y\|^2\|x\|^2 \leq 0
|(x,y)| \leq \|x\|\|y\|
which is Cauchy-Schwarz inequality.
Does that help?