Use Theorem 4.11 to prove that if \(\left\{X_{n}ight\}_{n=1}^{\infty}\) is a sequence of random variables that converge in distribution to a real constant \(c\) as \(n ightarrow \infty\), then \(X_{n} \xrightarrow{p} c\) as \(n ightarrow \infty\).

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Theorem 4.11 (Slutsky). Let {X} be a sequence of random variables that converge weakly to a random variable X. Let {Y}=1 be a sequence of random variables that converge in probability to a real constant c. Then, 1. Xn+Y X + c as n o. 2. X,Y, dcX as n . 3. Xn/Yn + X/c as n as long as c +0.