Is there any way to prove the Fourier Theorem ?
Any single valued periodic function can be represented by a summation of simple harmonic terms having frequencies which are the integral multiples of the frequency of the periodic function.
$f(t) = a_0 + \displaystyle \sum_{n=1}^{\infty} (a_n \cos n\omega t + b_n \sin n \omega t)$
Depends on what you mean by "represented by". If you mean the series actually converges to $f(t)$ for every $t$ then this can't be proved because it's false, even for continuous $f$.
But $f$ can nonetheless be recovered from its Fourier series. For example if $f$ is continuous and we define $$A_r(t)=a_0+\sum_{n=1}^{\infty}r^n (a_n \cos n\omega t + b_n \sin n \omega t)$$for $0<r<1$ then $A_r\to f$ uniformly as $r\to1$; one certainly might regard this as a sense in which $f$ is "represented by" its Fourier series. (The proof proceeds by showing that $A_r$ is the convolution $f*P_r$, where $P_r$ is the Poisson kernel.)