Proof about Separating Points, Vanishing, and Uniform Closure

I'm wondering if I'm getting stumped here because of my limited familiarity with complex numbers. So if anybody sees what's going on, if you could answer with a mild hint/nudge rather than a full answer, I would appreciate it.

The problem:

Let $\displaystyle K = $ the unit circle in the complex plane, and $\displaystyle \mathscr{A} $ the algebra of functions of the form $\displaystyle f(e^{i \theta}) = \displaystyle \sum^{N}_{n = 0} c_{n}e^{in \theta}$ for $\displaystyle \theta \in \mathbb{R}$. Show that $\displaystyle A$ separates points on $\displaystyle K$.

So my thinking: I want to take two points $\displaystyle e^{ia} \ne e^{ib}$ and show that there is a function in $\displaystyle A$ such that $\displaystyle f(e^{ia}) \ne f(e^{ib})$. Well, assuming $\displaystyle N \geq M$, then $\displaystyle \displaystyle \sum^{N}_{n = 0} c_{n}e^{ina} = \sum^{M}_{m = 0} c_{m}e^{ima} \Leftrightarrow c_{1}(e^{ia} - e^{ib}) + ... + c_{N - M + 1}e^{i(N - M)a} + ... + c_{N}e^{iNa} = 0$. So I want to show that this last equality fails.