I have the following question:
We have $(X_n)_{n},X$ a collection of real valued random variables which are defined in $(\Omega, F, \Bbb{P})$. And $f:(\Bbb{R},B(\Bbb{R}))\rightarrow (\Bbb{R},B(\Bbb{R}))$ a continuous function. I need to show that if $X_n\rightarrow X$ a.s. then $f(X_n)\rightarrow f(X)$ a.s.
So I mean if we assume $X_n\rightarrow X$ a.s. this means that there exists $N\in F$ s.t. $\Bbb{P}(N)=0$ and for all $\omega \in \Omega \setminus N$ $$X_n(\omega)\rightarrow X(\omega)$$ I thought maybe I can apply $f$ to $N$ but this confuses me a bit since $f$ is not neccessairly defined on $N$. Then I thought one could do it by contraposition but also there I got stuck.
Could maybe someone help me?
Let $\omega\in \{X_n\to X\}$. Then $f(X_n(\omega))\to f(X(\omega))$ by usual rules of limits. This means that $\{X_n \to X\}\subseteq \{f(X_n)\to f(X)\}$. But then $1=P(X_n \to X)\leq P(f(X_n)\to f(X))=1$. So $f(X_n) \to f(X)$ (a.s.). Note all these sets are measurable because $X$ is measurable and $f$ is continuous. In particular:
$$\{X_n \to X\}=\bigcap_{k \in \mathbb{N}}\bigcup_{N \in \mathbb{N}}\bigcap_{n \geq N}\{|X_n-X|<1/k\}$$