Prove that the normed space $L^{\infty}$ equipped with $\lVert\cdot\rVert_{\infty}$ is complete. [duplicate]

Solution 1:

Let $\left(f_n\right)_{n\geqslant 1}$ be a Cauchy sequence in $L^{\infty}$ endowed with the natural norm. For $n,m$, let $N_{n,m}$ a set of measure $0$ such that $|f_n(x)-f_m(x)|\leq \lVert f_n-f_m\rVert_{\infty}$ for each $x\notin N_{n,m}$. Define $N:=\bigcup_{n,m}N_{n,m}$. Then $N$ is of measure $0$ as a countable union of such sets and the sequence of functions $\left(\widetilde f_n\right)_{n\geqslant 1}$ restricted to $N^c$ is uniformly convergent. Then you can find an uniform limit $f$ (i.e. such that $\lVert f-\widetilde f_n\rVert_{\infty}\to 0$. Then just define $f$ by $0$ on $N$ to get a limit in $L^{\infty}$ (more precisely the limit will be the equivalence class of this function).