How to show that:
$\lim_{x\rightarrow 0} \frac{1}{x}\int_0^x |\sin(1/y)| \mathrm{d} y \not=0$
It seems like a easy example of illustrating 0 is not in the Lebesgue set of $g(x)$ where $g(x)=\sin(1/x)$ if $x\neq 0$ and $g(0)=0$. But I fail to see why the above integral is true.
I tried looking at the intervals such that $\sin(1/y)$ is greater or equal to some constant (for example, $\left[\frac{1}{k\pi+\pi/6}, \frac{1}{k\pi+5\pi/6}\right]$ such that $\sin(1/y)\geq \frac{1}{2}$), however, $\sum_{k \text{ large}} \left(\frac{1}{k\pi+\pi/6}-\frac{1}{k\pi+5\pi/6}\right)$ converges, which is not strong enough to prove the claim. Any thoughts? Thanks in advance.