Problem Statement: Suppose that $\{f_n\}_{n=1}^\infty$ is a sequence of continuous functions on $R$ converging to a function $f$ on $R$. Prove that there exists a non-empty open set $U$ of $R$ and a real number $M$ such that $|f_n(x)|\le M$ for all $x \in U$ and $n \ge 1$.
My attempt at a solution: I'm not sure how to proceed here - what I've done so far is: note that $f$ is measurable, since each continuous function is measurable, and messed around with sets of the form $E_M = \{x : |f_n(x)| \ge M \text{ for all } n\}$, trying to get some kind of intersection or union of those sets to be the set I want. But, I don't quite see how this should work, because I don't have that $f$ is integrable. I would really appreciate a hint in the right direction!