$L^\infty$ convergence does not imply uniform convergence almost everywhere

Question

It is true the following:

Proposition: If $\left\{u_k\right\}$ is a sequence of functions such that $u_k \to u$ in $L^p$, for $p \in [1, \infty)$, there exists a subsequence which converges almost everywhere.

Question: What could serve as a counterexample for the case $p= \infty$?

Answer 1

The proposition is true, by the following theorem:

If $1\leq p\leq\infty$ and if $\{f_n\}$ is a Cauchy sequence in $L^p(\mu)$, with limit $f$, then $\{f_n\}$ has a subsequence which converges pointwise almost everywhere to $f(x)$.

This is Theorem $3.12$ of Rudin's "Real and Complex Analysis", third edition.

Note that the theorem is also valid for $p=\infty$, so there is no counterexample.