Let $f(t)=(t-\pi)\chi_{(0,2\pi)}$, $t \in [0,2\pi]$, then the partial sum of the Fourier series of $f$ is $ S_n(t)=- \sum_{0 < |k| \le n} \frac{\sin k t}{k}. $ Show $|S_n(t)| \le \pi+2$ for all $n$ and $t$.
I know the Fourier series of $f$ converges to $f(t)$ on $(0,2\pi)$, to $-\pi/2$ at $0$, and to $\pi/2$ at $2\pi$.
So there is an $n_0$ past which I can uniformly bound $S_n(t)$ by $\pi+2$.
But I don't know how to handle the term before $n_0$.
I considered writing $ S_n(t)=\int_0^t \left(\frac{\sin (k+1/2) x}{\sin x/2} - 1 \right)dx $
But I don't see how to make $\pi+2$ appear.