Corollary of convergence in distribution 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.

Write each expectation as integral of the tail on the positive real line, and apply Fatou's lemma to the positive real line (with Lebesgue measure) and the sequence $f_n(t)=\mathbb P(X_n>t)$.
