Fourier series is a way to express a periodic function as a sum of complex exponentials or sines and cosines.. Is there actually a proof for the fact tat a periodic function can be split up into sines and cosines or complex exponentials?
It is fairly easy to show that any integrable periodic function can be approximated arbitrarily well by a sum of sines and cosines. The Fourier series is the limit of those approximations as the "error" goes to 0- except on a set of measure 0 for dis-continuous functions. I might point out that the other way is what's hard. It can be shown that some perfectly valid Fourier series converge to non-(Riemann)-integrable functions. That was why Lebesque integration had to be invented.