Prove the third result of Theorem 4.11. That is, let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables that converge weakly to a random variable \(X\). Let \(\left\{Y_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables that converge in probability to a real constant \(c eq 0\). Prove that \(X_{n} / Y_{n} \xrightarrow{d} X / c\) 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.