I don't understand this step in the proof about L infinity

[futurebird]

I'm learning the proof that $$L_{\infty}$$ is complete. I do not understand one of the steps.

Let $$f_n$$ be a cauchy sequence in $$L_{\infty}(E)$$ then there exists a subset A in E such that $$f_n$$ is "uniformly cauchy" on E\A. For m,n choose A so that

$$|f_n-f_m| \leq ||f_m - f_n||_{\infty}$$ for all x in E\A. Take the union of all such As and then $$f_n$$ converges uniformly on E without the As.

Define f to be $$f(x) = \mathop{\lim}\limits_{n \to \infty} f_n(x)$$ for x in E without the As, and let it be o otherwise. F is bounded and measurable now all we need is to show that $$||f_n - f||_{\infty} \rightarrow 0$$ so we know f is in $$L_{\infty}$$. We know $$m(As)=0$$

This next bit is where the proof makes a leap that I don't understand.

$$||f_n - f||_{\infty} \leq \sup_{x \in E-A} |f_n -f|$$

Then is says as n --> infinity we have $$\sup_{x \in E-A} |f_n -f| \rightarrow 0$$. But, I have no idea where that inequality came from? What theorem? Please help!

 

[futurebird]

*a little bump*

I know it's not a simple proof I'll see what my prof. says in office hours. But, hmmm maybe it has something to do with the l infinity norm being the esssup?

I'm still not used to the esssup and I have a test in 4 days...