I am looking for a proof of this fact: let $f$ be a $2\pi$-periodic function, $f\in L^1(-\pi,\pi),$ and let $a_k,b_k$ be its Fourier coefficients. If $\sum_{k=1}^\infty{|a_k|+|b_k|}<\infty,$ then the Fourier series of $f$ converges to $f.$
I know that the Fourier series of $f$ converges (uniformly) to a certain $g$ because of Weierstrass M-Test, but how can I prove that $g=f?$