I'm trying to show that any continuous function $f$ with period $2\pi$ can be approximated by a fourier series $P$ (i.e. $| P(x) - f(x)| < \epsilon $ for $\epsilon > 0$). Any suggestions?
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If it is uniform approximation that you want, then you can use Fejer's Theorem.
Here's a nice section of convergence of fourier series for piece-wise continuous functions: http://www.sosmath.com/fourier/fourier3/fourier3.html. Have a read through, and let us know if you need more help!