Showing $|\sum_{k=1}^n\frac{\sin(kx)}k|

Showing $|\sum_{k=1}^n\frac{\sin(kx)}k|

For any real $x$ and positive integer $n$, is it true that:
$$\left|\sum_{k=1}^n\frac{\sin(kx)}k\right|<2\sqrt{\pi}\quad ?$$
Please justify.