Suppose that $(f_n)$ is a sequence of functions for which there exists a finite constant $C$ such that the $L_\infty$ norm of $f_n$ is less than or equal to $C$ for all $n$. Suppose further that $f_n$ converges "almost uniformly" to a function $f$, i.e. for every $\epsilon>0$ there exists a set $E$ whose complement has measure less than $\epsilon$ such that $f_n$ converges uniformly to $f$ on $E$. Then show that $f$ is in $L_\infty$.
I'm not sure how to approach this. One thought I had is using the result that $L_\infty$ convergence is equivalent to uniform convergences almost everywhere, i.e. there exists a set $E$ whose complement has measure $0$ such that $f_n$ converges to $f$ uniformly on $E$. But I'm not sure how to use the boundedness of the $L_\infty$ norms and almost uniform convergence to prove uniform convergence almost everywhere.