Is $\frac{\sin(x)}{x}$ Lebesgue-Integrable?

Question

I'm trying to understand why $\frac{\sin(x)}{x}$ is not integrable according to Lebesgue over $\mathbb{R}$

If found this answer that helped me a lot but don't understand how the Monotone Convergence Theorem was used

https://math.stackexchange.com/a/3184570/741674

Can someone help me by detailing the problematic line in this answer?

Thank you

Answer 1

The author is applying the MCT to the non-decreasing sequence of functions $$f_N(x)=\sum_{k=0}^N\Bbb{1}_{[{2k\pi},\, {(2k+1)\pi}]}(x)\frac{\sin(x)}x $$ where $$\Bbb 1_S(x)=\cases{1 & if $x \in S$,\\0 & otherwise.}$$