Let $(\Omega$, $\mathcal{F}$, $P)$ be a probability space, $X_n$, $X$ random variables in that space. Show that:
$(a)$ $\{\omega\in\Omega| \lim_{n\rightarrow\infty} X_n(\omega)$ $exists\}\in\mathcal{F}$,
$(b)$ $\{\omega\in\Omega| \lim_{n\rightarrow\infty} X_n(\omega)=X(\omega)\}\in\mathcal{F}$.
Edit:
I found quite simply solution.
$\{\omega\in\Omega| \lim_{n\rightarrow\infty} X_n(\omega)exists\}=\\=\{\omega\in\Omega| \forall_{k\in\mathbb{N}}\exists_{N\in\mathbb{N}}\forall_{m,n>N}|X_n(\omega)-X_m(\omega)|<\frac1k\}= \\=\bigcap\limits_{k\in\mathbb{N}}\bigcup\limits_{N\in\mathbb{N}}\bigcap\limits_{n,m>N}\{|X_n(\omega)-X_m(\omega)|<\frac1k\}=\\=\bigcap\limits_{k\in\mathbb{N}}\bigcup\limits_{N\in\mathbb{N}}\bigcap\limits_{n,m>N}\big(\{X_n(\omega)-X_m(\omega)<\frac1k\}\cap\{X_n(\omega)-X_m(\omega)>-\frac1k\}\big)\in\mathcal{F}$