Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 26L / c / Solution

Yes. The $L^2$ bound makes the family uniformly integrable in $L^1$: $E[|X_n|1_{|X_n|>K}]\le E[X_n^2]/K\le 1/K$. Together with convergence in probability, Vitali's theorem gives $E|X_n|\to 0$.

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