Let $V$ be a vector space over either $\mathbb{R}$ or $\mathbb{C}$ Let $\langle.,.\rangle$ be an inner product on $V$.
Prove the Cauchy Schwarz Inequality
$$\forall x,y\in V, |\langle x,y\rangle|^2 \leq |\langle x,x\rangle||\langle y,y\rangle|$$
all the proofs I can come up with right now involves some sort of norm on $V$. I don't think I can just assume that $V$ has a norm.
So are there ways to prove it using nothing but the definition of inner product?