Statistic convergence as surely question

Question

Suppose $X_n$ converges almost surely to $X$, and $f$ is a continuous function. Prove that $f(X_n)$ converges almost surely to $f(X)$.

My approach: according to definition of continuous $|x - a| < \alpha$, then $|f(x) - f(a)| < \delta$ so that take the limit of n to infinity: $P(|f(X_n) - f(X)| < \delta) = 1$ but I am not sure what to do after that?

Can someone please help me out?

(homework)(convergence as surely)

Answer 1

That $X_n\to X$ almost surely means that there exists a null-set $N$ such that $$ X_n(\omega)\to X(\omega),\quad \text{for all }\,\omega\in N^c. $$ Since $f$ is continuous we know that $f(a_n)\to f(a)$ for every convergent sequence $(a_n)_{n\geq 1}$ converging to $a$. Try to use this to conclude that $f(X_n)\to f(X)$ almost surely.