Let $(T,\mathcal{A})$ be a $\sigma$-algebra, and $(f_{n})$ a sequence of $\mathcal{A}$-measurable functions from $T$ to $\mathbb{R}$. Show that the set $\{ t \in T: \lim_{n\rightarrow \infty}f_{n}(t)\text{ exists and is finite}\}$ is in $\mathcal{A}$.
I try with the definition of pointwise convergence, that is, $\forall \epsilon >0$, $\exists n_{0}$ such that $|f_{n}(t)-f(t)|<\epsilon$, now $f_{n}(t) \in (f(t)-\epsilon,f(t)+\epsilon)$ and like $f_{n}$ is measurable the set is in $\mathcal{A}$. Can I use this to resolve the problem, or does there exist another way?
Any help is appreciated.