Almost Sure convergence for sequence of random variables

Question

Let $X_n$ be a sequence of random variables converging to $X$ almost surely. $f:{\rm I\!R} \to {\rm I\!R}$ is a continuous function. Then, $f(X_n)$ converges to $f(X)$ almost surely.

Answer 1

I need to show that $P\{ w:f(X_n)(w) \to f(X)(w)\} = 1$

I know that $P\{ w:X_n(w) \to X(w)\} = 1$

Let $w$ be such that $X_n(w)\to X(w)$

Then because $f$ is continuous $f(X_n(w))\to f(X(w))$

so $\{ w:X_n(w) \to X(w)\}$ $\subseteq$ $\{ w:f(X_n)(w) \to f(X)(w)\}$

The smaller set has probability 1 so the bigger set must also have probability 1.