Let $\{f_n\}$ be a sequence of functions in $L^\infty$. I want to prove that $\{f_n\}$ converges to $f\in L^\infty$ $\Leftrightarrow$ there is a set $E$ of measure zero such that $f_n$ converges uniformly to $f$ on $E^c$.
My Attempt:
$(\Rightarrow)$ Suppose $f_n\rightarrow f$ in $L^\infty$. Then $ \Vert f_n-f\Vert_\infty=\inf \{M:|f_n-f|\leqslant M ~~\text{a.e.}\}\lt \varepsilon$ for any $\varepsilon \gt 0$. Let $E=\{x:|f_n(x)-f(x)|\gt \varepsilon\}.$ The $m(E)=0$. So $f_n\rightarrow f$ uniformly on $E^c$.
$(\Leftarrow)$ Suppose $f_n\rightarrow f$ uniformly on $E^c$ with $m(E)=0$. Then $\forall~t\in E^c$ and $n\gt N$, $|f_n(t)-f(t)|\lt \varepsilon$. But
$|f(t)|=|f(t)-f_n(t)+f_n(t)|\leqslant |f(t)-f_n(t)|+|f_n(t)|\lt \varepsilon + \Vert f_n(t)\Vert_\infty.$ So $f$ is bounded and hence $f\in L^\infty$.
Please, could someone look over what I've done and point out any errors?
Thanks.