Let $X,X_1,X_2,...$ be real valued r.v. s.t. $\mathbb{E}[Uf(X_n)] \rightarrow \mathbb{E}[Uf(X)]$ for all bounded r.v. U and $f \in C_b(\mathbb{R},\mathbb{R})$ and let $\sup_n\mathbb{E}[(X_n)^p]<\infty$ for all $p \geq 1$. Is it true that
$$\mathbb{E}[(X_n)^2] \rightarrow \mathbb{E}[X^2]$$
holds? I think the paper I am reading uses this, but I don't know why this should hold. I would be grateful for hints or a solution.
Notice that taking $U=1$ gives that $X_n \to X$ in distribution. Using Skorokhod's representation theorem, we can assume that there is almost sure convergence.
Now since $X_n$ is bounded in every $L^p$, $X_n^2$ is uniformly integrable. Thus Vitali's theorem gives the convergence $X_n \to X$ in $L^2$.