If I have a set $E_n = \{x \in E : |f(x)| < n \}$, then what is $\bigcup_{n}^\infty E_n$ and $\bigcap_{n}^\infty E_n$?
The set $\bigcup_{n=1}^\infty\{x\in E\mid|f(x)|<n\}$ is simply $E$, since, for each $x$, you have $|f(x)|<n$ for some $n$ (and, in fact, for infinitely many $n$'s).