Let $K$ be the unit circle in the complex plane and let $A$ be the algebra of all functions of the form $f(e^{i\theta})=\Sigma_{n=0}^{N}c_{n}e^{in\theta}$ ($\theta$ real). Prove that $A$ is uniformly dense in the space of continuous functions from unit circle to complex plane.
This is a problem from Rudin. The things I have in mind for this is that I need to prove the following: 1) $A$ separates points in $K$. 2) $A$ vanishes at no point of $K$.
The solution manual says that : Note that $f(e^{i\theta})=e^{i\theta}\in A$ and for every $f\in A$, $\int_{0}^{2\pi}f(e^{i\theta})e^{i\theta}d\theta=0$.
How do we prove these two things? the solution manual is not that obvious to me. I am reading Rudin on my own. please explain.