I'd really love to understand why does the fact that the series of the absolute Fourier coefficient converges ($\sum_{n \in \mathbb{Z}}|\widehat{f(n)}|<\infty$) for a function $f$, leads to the fact that the whole Fourier series, $\sum_{n \in \mathbb{Z}}\widehat{f(n)}e^{inx}$, uniformly converges.
Thanks a lot.