Corollary of convergence in distribution

Question

Can I say that if $\xi_n \xrightarrow {d} \xi$ and $\xi_n, \xi$ are non-negative random variables with finite expected value then
$$\mathbb{E}\xi \le \lim{\inf{}_{n\to\infty}\mathbb{E}\xi_n}?$$
I have problems to disprove or prove it. May somebody can help me? I would be grateful.

Answer 1

Using Skorohod's representation theorem, there exists a probability space $(\Omega, \mathcal F, \mathbb P)$, and random variables $X_n,X$ on that space, such that $X_n\stackrel{d}= \xi_n$, $X\stackrel{d}= \xi$, and $X_n\xrightarrow{\text{a.s.}} X$. Conclude using Fatou's lemma that $$E\xi =EX\le \liminf_n EX_n=\liminf_n E\xi_n.$$