Prove the first part of Theorem 3.7. That is let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables, \(c\) be a real constant, and \(g\) be a Borel function on \(\mathbb{R}\) that is continuous at \(c\). Prove that if \(X_{n} \xrightarrow{a . c .} c\) as \(n ightarrow \infty\), then \(g\left(X_{n}ight) \xrightarrow{\text { a.c. }} g(c)\) as \(n ightarrow \infty\).

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Theorem 3.7. Let {X} be a sequence of random variables, c be a real constant, and g be a Borel function that is continuous as c. a.c. 1. If Xn cas no, then g(Xn) 2. If Xnc as n , then g(Xn) a.c. g(c) as n . g(c) as n 0.