A question concerning a set connected to a sequence of measurable functions I came across the following problem in my study of measurable functions:
If $\{f_n\}$ is a sequence of measurable functions on $X$, then $\{x : \lim_{n \to \infty}f_n(x) \mbox{ exists}\}$ is measurable.
I know how to prove this result if we assume the $f_n$ are $\overline{\mathbb{R}}$-valued, since there are particularly nice results about the limit inferior and the limit superior of such functions in this case. I suspect these (lim sup and lim inf) will factor into the more general case, but I was wondering how the proof would proceed!

Expanding on Srivastans comment: Let $C$ be the set of points $x$ where $(f_n(x))$ converges. This is the case if and only if $(f_n(x))$ is Cauchy: $$C=\big\{x:\forall\epsilon>0 \exists N\forall m> N\forall n>N:|f_n(x)-f_m(x)|<\epsilon\big\}$$
$$C=\big\{x:\forall k \exists N\forall m>N\forall n>N:|f_n(x)-f_m(x)|<1/k\big\}$$
$$C=\bigcap_{k=1}^\infty\bigcup_{N=1}^\infty\bigcap_{n=N+1}^\infty\bigcap_{m=N+1}^\infty\{x:|f_n(x)-f_m(x)|<1/k\}$$
