I'm looking for an elementary way of showing the following. If $(X_n)$ and $X$ are random variables such that $X_n \to X$ in distribution and such that $\{X_n\mid n\geq 1\}$ are uniformly integrable, then $E[X_n]\to E[X]$.
I've seen another topic on this, but the solution given there is using Skorokhod's theorem stating that convergence in distribution is equivalent to almost-sure convergence of copies of the random variables in some abstract probability space. I would like to do without that if possible. Thanks in advance!