TITLE: How to prove limit of measurable functions is measurable
QUESTION [32 upvotes]: I need help to prove the following theorem 

Suppose $f$ is the pointwise limit of a sequence of $f_n$, $n = 1, 2, \cdots$, where $f_n$ is a Borel measurable function on $X$. Then $f$ is Borel measurable on $X$.

My idea is to use the standard definition like for every $c$,$\{x:f(x)<c\}$ is Borel measurable. But got stuck as how to do it for sequence of $f_n$.

REPLY [10 votes]: As a more "direct" proof, you could also note that
$$\{x\in X\mid f(x)>a\}=\bigcup_{m=1}^\infty\bigcup_{N=1}^\infty \bigcap_{n=N}^\infty \left\lbrace x\in X :f_n(x)>a+\frac{1}{m}\right\rbrace$$