I found a simple proof for the completeness of $L^\infty$, based on the fact that a normed vector space is complete if $$\sum_n ||f_n||<\infty \implies \exists y=\sum_n f_n$$ I am not sure that the last passages work.
Let $f_n$ be a sequence in $L^\infty(\Omega,\mu)$ such that $$\sum_n ||f_n||_\infty=M<\infty$$ then by definition of ess sup exists $\{\Omega_n\}_n$ such that $\mu(\Omega_n^c)=0$ for all $n$ and $$\sup_{\Omega_n} f_n=||f_n||_\infty$$ then,
since $\mu((\cap_n \Omega_n)^c)=\mu(\cup_n \Omega_n^c)=0$, we have $$||\sum_n f_n||_\infty\le \sum_n ||f_n||_\infty<M$$ So $\sum_n f_n\in L^\infty$
Does the formulas in the rectangle hold?