Here is the question:
Let $\xi_n $ be a sequence of random variables on a probability space $(\Omega,\mathcal{F},P) $ such that $E \xi_n^2 \le c $ for some constant $c$. Assume that $\xi_n \to \xi $ almost surely as $n \to \infty$. Prove that $E \xi $ is finite and $E \xi_n \to E \xi $.
I guess the condition that $ E \xi_n^2 \le c$ is really important here. But I don't know how to use it correctly.
Thanks to the suggestion of Did, I'm able to answer this question now. The answer is as follows.
$E \xi_n^2 \le c$ means that $\{ \xi_n \} $ is uniformly bounded in $L^2(\Omega,\mathcal{F},P)$, which suggests that $\{ \xi_n \}$ is uniformly integrable.
$E\xi_n^2 \le c$ implies that $E |\xi_n| \le \sqrt{c}$, which further implies $|\xi_n | < \infty$ almost surely. Since $\xi_n \to \xi$ almost surely, we are able to say that $|\xi| < \infty$ almost surely.
Now, we can use Vitali convergence theorem, which implies that $E \xi$ is finite and $E |\xi_n - \xi| \to 0 $ as $n \to \infty $.