Question is to prove that : 

Set of all points at which a sequence of measurable functions converge is a measurable set..

What i have tried is as follows :

We are looking at the following set : 

$$\{x\in X : (f\_n(x)) \text{converges}\}$$

Which is same as 
$$\{x\in X : (f\_n(x)) \text{is cauchy}\}$$

Which is same as 

$$\{x\in X : \text{given $\epsilon>0$ there exists $N\in \mathbb{N}$ such that $|f\_n(x)-f\_m(x)|<\epsilon$ for all $m,n\geq N$}\}$$

I want to write this as unions and intersections of measurable sets and then conlcude this is measurable.. Something like :

$$\bigcap\_{p\in \mathbb{R}??}\bigcup\_{m,n\in \mathbb{N}??}\{x:|(f\_n-f\_m)(x)|<p\}$$

Now, As $f\_n,f\_m$ are measurable so is $f\_n-f\_m$ and so is $|f\_n-f\_m|$ and $\{x:|(f\_n-f\_m)(x)|<p\}$ being inverse image of open set is measurable..

I am not so sure how to write that as unions and intersections...