Question: Use the monotone convergence theorem to prove the following.

1. If \(X_{n} \geq 0, X_{n} \downarrow X\) a.s. and \(E X_{n}<\infty\) for some \(n\) then \(E X_{n} \rightarrow E X\).
2. If \(E|X|<\infty\) then \(E|X| 1_{(|X|>n)} \rightarrow 0\) as \(n \rightarrow \infty\)
3. If \(E\left|X_{1}\right|<\infty\) and \(X_{n} \uparrow X\) a.s. then either \(E X_{n} \uparrow E X<\infty\) or else \(E X_{n} \uparrow \infty\) and \(E|X|=\infty\)
4. If \(X\) takes values in the non-negative integers then

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