Rietz-Fischer Theorem: Let $E$ be a measurable set and $1 \le p \le \infty$. Then every rapidly Cauchy sequence in $L^p(E)$ converges both with respect to the $p$-norm and pointwise almost everyone on $E$ to a function in $L^p(E)$.
Question: How does one prove this in the case that $p = \infty$? Every proof I have seen thus far seems to assume that $p \in [1,\infty)$. For example, the wikipedia proof makes this assumption and then concludes by remarking "the case $p = \infty$ reduces to a simple question about uniform convergence". Upon thinking about this for a bit it's not obvious how to proceed.
