Given a measurable space $(X,\scr{A})$ and a sequence of measurable functions $(f_n)_{n=1}^{\infty}$ where $f_n:X \to [-\infty,\infty]$, I would like to show that the set $B = \{x \in X \:{:}\:|\lim_{n \to \infty}f_n(x)| < \infty\}$ is measurable.
I know $[f_n>-\infty] \in \scr{A}$ is true since $f_n$ is measurable, but do not know how to extend this idea to show $B$ is measurable. Any help will be greatly appreciated.
Note that $x\in B\iff(f_n(x))_n$ is a Cauchy sequence, meaning exactly that:$$\forall n\in\mathbb N\exists m\in\mathbb N\forall k,l\geq m|f_k(x)-f_l(x)|\leq\frac1n$$
So $$B=\bigcap_{n\in\mathbb N}\bigcup_{m\in\mathbb N}\bigcap_{k,l\geq m}\{x: |f_k(x)-f_l(x)|\leq\frac1n\}$$
The sets $\{x: |f_k(x)-f_l(x)|\leq\frac1n\}$ are measurable and the collection of measurable sets is closed under countable intersections and countable unions.