I am finding an alternative proof of the following theorem:
Th. Let $(X,\mathcal S)$ be a measurable space.Let $(f_n)$ be a sequence of measurable functions from $X\to \mathbb R$.If $f=\lim f_n$ exists then $f$ is measurable.
I am looking for an easy and intuitive proof rather than the one I found in my textbook by Axler:
Let $a\in \mathbb R$.
It suffices to show that $f^{-1}(a,\infty)\in \mathcal S$.
Now,$f^{-1}(a,\infty)=\bigcup\limits_{j=1}^\infty\bigcup\limits_{m=1}^\infty\bigcap\limits_{k=m}^\infty f_k^{-1}(a+\frac{1}{j},\infty)$ and from here the claim follows.
But the problem with this proof is that it is not intuitive although it is a short one.Can someone give a different proof?