I'm trying to understand the proof below. 

Why was the set $N\_{n,m}$ of measure zero defined in this proof?

Why is the sequence uniformly convergent?

[Prove that the normed space $L^{\infty}$ equipped with $\lVert\cdot\rVert\_{\infty}$ is complete.](https://math.stackexchange.com/questions/150306/prove-that-the-normed-space-l-infty-equipped-with-lvert-cdot-rvert-inft)

[![enter image description here](https://i.stack.imgur.com/a8cJk.png)](https://i.stack.imgur.com/a8cJk.png)