Absolute convegence of Fourier series implies uniform convergence? I've read many statements in my textbook that include something along the lines of "the Fourier series of $f$ converges absolutely, and hence uniformly to $f$". I'm not sure why this is true. 
If the Fourier series converges absolutely, then it converges. That is, $ \lim_{n \to \infty} S_n(\theta) $ converges for each $\theta$. Why does it follow then that this limit is uniform?

Let $$S_n(\theta)=\sum_{|k|\leq n}c_ke^{2i\pi k\theta}$$ be the partial Fourier series.
Now, suppose that for all $\theta \in [0, 2\pi]$, we have absolute convergence $S_n(\theta)\rightarrow f(\theta)$.
Let $\varepsilon>0$. By definition of the series being absolutely convergent, $$\sum_{k\in\mathbb Z}|c_k|<+\infty$$
Therefore, the partial sums are Cauchy:
Let $\varepsilon>0$, there exists $N>0$ such that if $n>N$ and $m>N$, $$\sum_{|k|>n,m}|c_k|<\varepsilon$$
So for all $\theta \in [0, 2\pi]$,
$$\left|S_n(\theta)-S_m(\theta)\right| \leq \sum_{|k|>n,m}|c_k| < \varepsilon$$
So $\{S_n\}_{n\in\mathbb N}$ is uniformly Cauchy, so it converges uniformly.
