Let $\displaystyle (f_n)$ be a sequence of real-valued measurable functions on $\displaystyle [a,b]$. If $\displaystyle (f_n)$ converges to a function $\displaystyle f$ almost everywhere on $\displaystyle [a,b]$, prove that there exists a sequence $\displaystyle (E_n)$ of measurable subsets of $\displaystyle [a,b]$ such that

$\displaystyle m([a,b] \setminus \cup E_n) = 0$

and $\displaystyle (f_n)$ converges uniformly to $\displaystyle f$ on each $\displaystyle E_n$.

How do I even start this question?

Thanks in advance.