Edit The question as originally phrased was clumsy. What I really need is the simplest proof, or reference, anyone can rustle up of this: "for $G$ a compact Lie group, and $g$ and $h$ distinct elements of $G$, then there's a finite-dimensional complex representation $\rho$ with $\rho(g)$ not equal to $\rho(h)$."
Once this is in hand, the Stone-Weierstrass theorem quickly gives me one version of Peter-Weyl, and from that I think I can deduce all the versions I care about!
I've been looking around for an easyish proof of this online, but every source I find seems to want to prove something for compact, not necessarily Lie, groups, and jumps in with the theory of compact operators and so on - naively I hope there's an easier way if one only wants to know about Lie groups...