Understanding $\sum^n_{i=1}\int_{(i-1)\pi}^{i\pi}|\frac{\sin x}{x}|\,dx\ge\sum^n_{i=1}\frac{1}{i\pi}\int_{(i-1)\pi}^{i\pi}|\sin x|\,dx$

Question

Could some please explain me the following inequality?

$$\sum^n_{i=1} \int_{(i-1)\pi}^{i\pi} \left|{\frac{\sin x}{x}}\right| \, dx\geq \sum^n_{i=1}\frac{1}{i\pi}\int_{(i-1)\pi}^{i\pi}\left|\sin x\right| \, dx $$

Thank you!

Answer 1

For $i=1,\ldots,n$ and $(i-1)\pi\leq x\leq i\pi$ $$ \left|\frac{\sin x}{x}\right|=\frac{|\sin x|}{x}\geq \frac{1}{i\pi}\left|\sin x\right| $$