Sequence of Measurable Functions defined on R

I'm struggling with this problem from my most recent homework assignment in measure theory.

Let {$f_n$} be a sequence of measurable functions defined on $\Bbb R.$ Show that the set

$E =$ {$x \in \Bbb R: \lim_{n \rightarrow \infty} f_n(x)=\infty$} is measurable.

There is an identical problem for $- \infty$ as well, but it's really more about the concepts and method that I'm struggling with. I don't know how to go about showing this set is measurable.

{$x \in \Bbb R: \lim_{n \rightarrow \infty} f_n(x)=\infty$} can be rewritten as
$\{x \in \Bbb R: \forall N \in {\mathbb N}$ $\exists M \in {\mathbb N}$ such that if $n \geq M$ then $f_n(x) > N\}$
This is the same as $$\cap_{N = 1}^{\infty} \cup_{M = 1}^{\infty} \cap_{n =M}^{\infty}\{x \in {\mathbb R}: f_n(x) > N\}$$
Since each $\{x \in {\mathbb R}: f_n(x) > M\}$ is measurable so is {$x \in \Bbb R: \lim_{n \rightarrow \infty} f_n(x)=\infty$}. Basically, the idea is you can write such a set using $\forall$s and $\exists$s and then you can turn those into intersections and unions like the above.