Prove $\forall n \in \mathbb{N}: \int_{0}^{\frac{\pi}{2}} |\frac{\sin(nx)}{x}|dx \geq \frac{2}{\pi}\sum_{k=1}^{n}\frac{1}{k}$ As my further preparation to Putnam competition, I came across such inequality to prove:
$$\forall n \in \mathbb{N}: \int_{0}^{\pi} \left|\frac{\sin(nx)}{x}\right|dx \geq \frac{2}{\pi}\sum_{k=1}^{n}\frac{1}{k}$$
The problem is that I got stuck in spite of spending by far 6 days on this problem!
Because I got really stuck, I am very determined to see how to prove such inequality. Help very, very appreciated!
 A: We may start from:
$$ A_k = \int_{0}^{\pi}\frac{\sin x}{x+k\pi}\,dx = \int_{0}^{\pi/2}\sin(x)\left(\frac{1}{x+k\pi}+\frac{1}{\pi-x+k\pi}\right)\,dx$$
and notice that $f_k(x)=\frac{1}{x+k\pi}+\frac{1}{\pi-x+k\pi}$ is decreasing on $\left[0,\frac{\pi}{2}\right]$, hence:
$$ A_k \geq \int_{0}^{\pi/2} \sin(x)\,f_k\left(\frac{\pi}{2}\right)\,dx = \frac{4}{(2k+1)\pi}\geq \frac{4}{(2k+2)\pi}=\frac{2}{\pi}\cdot\frac{1}{k+1}.$$
Then we may notice that:
$$ \int_{0}^{\pi}\left|\frac{\sin(nx)}{x}\right|\,dx = \sum_{k=0}^{n-1}A_k \geq \color{red}{\frac{2}{\pi} H_n} $$
as wanted. My approach indeed proves a slightly stronger inequality, i.e.:

$$ \int_{0}^{\pi}\left|\frac{\sin(nx)}{x}\right|\,dx \geq \frac{2}{\pi}\sum_{k=1}^{n}\frac{1}{k\color{red}{-\frac{1}{2}}}.$$

A: We have $$\begin{align}
\int_{0}^{\pi}\left|\frac{\sin\left(nx\right)}{x}\right|dx\stackrel{nx=v}{=} & \int_{0}^{n\pi}\left|\frac{\sin\left(v\right)}{v}\right|dv \\
 = & \sum_{k=0}^{n-1}\int_{k\pi}^{\left(k+1\right)\pi}\left|\frac{\sin\left(v\right)}{v}\right|dv \\
 = & \sum_{k=0}^{n-1}\left(-1\right)^{k}\int_{k\pi}^{\left(k+1\right)\pi}\frac{\sin\left(v\right)}{v}dv \\
 \geq & \frac{1}{\pi}\sum_{k=0}^{n-1}\frac{\left(-1\right)^{k}}{k+1}\int_{k\pi}^{\left(k+1\right)\pi}\sin\left(v\right)dv \\ = &
 \frac{1}{\pi}\sum_{k=0}^{n-1}\frac{\left(-1\right)^{k}\left(-\cos\left(\left(k+1\right)\pi\right)+\cos\left(k\pi\right)\right)}{k+1} \\
 = & \frac{2}{\pi}\sum_{k=0}^{n-1}\frac{1}{k+1}.
\end{align}$$
Note that the third line is positive since $\sin\left(x\right)<0
 $ if $x\in\left(k\pi,\left(k+1\right)\pi\right)
 $ and $k$ odd. And $$-\cos\left(\left(k+1\right)\pi\right)+\cos\left(k\pi\right)=2\left(-1\right)^{k}.
 $$
A: Hint: 
$$\int_{(k-1)\pi}^{k\pi} \left| \frac{\sin x}{x} \right|\,dx \geq \int_{(k-1)\pi}^{k \pi} \frac{|\sin x|}{k \pi}$$
