This is Rudin’s RCA book. He discusses completeness of $L^infty(mu)$, but there is a part of proof that I don’t understand.

In $L^infty(mu)$, suppose ${f_n}$ is a Cauchy sequence, and let $A_k, B_{m,n}$ be the sets where $|f_k(x)| > |f_k|_infty$ and $f_n(x)-f_m(x)| > |f_n-f_m|_infty$, and let $E$ be the union of these sets, for $k, m ,n = 1,2,3, cdots$. Then, $mu(E)=0$, and on the completement of $E$, the sequence ${f_n}$ converges uniformly to a bounded function $f$. Define $f(x) = 0$ on $E$. Then, $f in L^infty(mu)$ and $|f_n – f|_infty rightarrow 0$ as $n rightarrow infty$.

Since $f$ is a complex function, I understand that $f_k$ converges to some measurable complex function $f$. The part that I do not understand is 1. (almost everywhere) boundedness, and 2. (a.e.) uniform convergence. Any help would be appreciated.