I'm new at measure theory and convergence, so bare with me.
I have some trouble with the following exercise:
"Let $X_1,X_2,...$ be random variables on $(\Omega,\mathcal{F},P)$. Show that the set $A=\{\omega; X_n(\omega) \text{ converges} \}$ is in $\mathcal{F}$ and that there exists a $\mathcal{F}$-measurable random variable $X$ such that $X_n(\omega) \rightarrow X(\omega)$ for $\omega \in A$.
Hint: Note that $A$ is the set such that all $k \in \mathbb{N}$, there exists an $n \in \mathbb{N}$ such that for all $m \in \mathbb{N}$ such that $|X_{n+m}-X_n|\lt 1/k$."
How do I start to think about this one? I'm not sure how to interpret the hint either.
If anyone could help me through this one I would be most grateful. Thank you in advance!
See here for a proof that $A$ is measurable.
Further you can define: $$X(\omega):=\lim_{n\to\infty}Y_n(\omega)\tag1$$ where $Y_n:=X_n\mathbf1_A$.
The $Y_n$ are measurable as product of two measurable functions and based on $(1)$ it can be shown that $X$ is measurable.
This evidently with $X(\omega):=\lim_{n\to\infty}X_n(\omega)$ for every $\omega\in A$.