Solution
= Solution
\b[Yes.] The $L^2$ bound makes the family uniformly integrable in $L^1$: $E[|X_n|1_{|X_n|>K}]\le E[X_n^2]/K\le1/K$. Together with convergence in probability, Vitali's theorem gives $E|X_n|\to0$.
= Solution
\b[Yes.] The $L^2$ bound makes the family uniformly integrable in $L^1$: $E[|X_n|1_{|X_n|>K}]\le E[X_n^2]/K\le1/K$. Together with convergence in probability, Vitali's theorem gives $E|X_n|\to0$.