# Prove set where limit of sequence of measurable functions exists is F-measurable

I'm working through Koralov's book on probability on my own and having some difficulty with problem 3.9.1:

Let $f_n$ and $f$ be measurable functions on a measurable space $( \Omega, \mathcal{F})$. Prove that the set $\{ \omega : \lim_{n \to \infty}f_n(\omega) = f(\omega) \}$ is $\mathcal{F}$-measurable. Prove that the set $\{ \omega : \lim_{n \to \infty}f_n(\omega) exists \}$ is $\mathcal{F}$-measurable.

I don't know the context for sure, but it seems safe to assume these functions are real-valued (or maybe complex-valued). However, you definitely can't assume $(f_n)$ is Cauchy in any sense. For any particular value of $\omega$, the sequence $(f_n(\omega))$ may not converge at all. The problem is asking you to think about the set of all $\omega$ such that it does converge (in the first question, to $f(\omega)$, and in the second question, to anything). For $\omega$ such that $(f_n(\omega))$ converges it will be a Cauchy sequence, but you don't know that it actually ever does converge.