Show that,$\int_0^\pi \left|\frac{\sin nx}{x}\right|\mathrm{d}x \ge \frac{2}{\pi}\left(1+\frac12+\cdots+\frac{1}{n}\right)$
$$ \begin{align} I &=\sum_{i=0}^{n-1} \int_{i\pi/n}^{(i+1)\pi/n} \frac{|\sin{nx}|}{|x|}\,dx \\ &=\sum_{i=0}^{n-1} \int_{0}^{\pi/n} \frac{|\sin{nx}|}{i\pi/n+ x}\,dx \\ &\gt \sum_{i=1}^{n} \int_{0}^{\pi/n} \frac{|\sin{nx}|}{i\pi/n}\,dx \\ &= \sum_{i=1}^{n} \frac{2/n}{i\pi/n} \\ &= \frac{2}{\pi}\sum_{i=1}^{n} \frac{1}{i} \end{align} $$