Prove the second part of Theorem 3.8. That is let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables, \(X\) be a random variable, and \(g\) be a Borel function on \(\mathbb{R}\). Let \(C(g)\) be the set of continuity points of \(g\) and suppose that \(P[X \in\) \(C(g)]=1\). Prove that if \(X_{n} \xrightarrow{p} X\) as \(n ightarrow \infty\), then \(g\left(X_{n}ight) \xrightarrow{p} g(X)\) as \(n ightarrow \infty\). Hint: Prove by contradiction using Theorem 3.5.

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Theorem 3.5. Let {X} be a sequence of random variables that converge in probability to a random variable X. Then there exists a non-decreasing sequence of positive integers {n} such that Xn X and Xnk X as k . a.c.