I've tried dotting around this article to little avail:
Any guidance?
2025-01-13 00:00:23.1736726423
Dominant and Monotone Convergence With Expectation
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Hint: $|X-X_n| = X_n-X +2(X-X_n)^+$, and the non-negative random variable $(X-X_n)^+$ is dominated by $X$. (Here $b^+:=\max(b,0)$.) I'm guessing that you are to assume that $\Bbb E[X]<\infty$.