Let $f$ be a Riemann integrable function on $[-\pi, \pi]$ such that $|\hat{f}(n)|\le \frac{K}{|n|}$ for some constant $K > 0$ and all $n\neq 0$. Show that $$|S_N(f)(x)|\le \sup_{y\in [-\pi, \pi]}|f(y)|+ 2K, \,\forall x \in [-\pi, \pi]$$
I think $\sup_{y\in [-\pi, \pi]}|f(y)|$ is used to control $\hat{f}(0)$, thus I need to show $|\sum_{n=-N,n\neq 0}^N \hat{f}(n)e^{inx}|\le 2K$. But I am stuck here.
PS: I think here $\hat{f}(n)$ means the $n$th Fourier coefficient, i.e. $\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\, dx$, since the question doesn't involve Fourier transform explicitly.