Lebesgue integral & Mathematical expectation

Let $\displaystyle \xi_1, \xi_2, ...$ a sequence of random variables on a probability space $\displaystyle (\Omega, \mathcal{F}, \mathbb{P})$ such that $\displaystyle \mathbb{E}\xi_n^2\leq c$ for some constant $\displaystyle c$. Assume that $\displaystyle \xi_n \rightarrow \xi$ almost surely as $\displaystyle n\rightarrow \infty$. How do you prove that $\displaystyle \mathbb{E}\xi$ is finite and $\displaystyle \mathbb{E}\xi_n\rightarrow\mathbb{E}\xi$?

I obviously thought about using the Dominated Convergence Theorem. Cauchy-Schwarz inequality ensures the expectation is finite. The problem is the sequence of expectations is dominated by a constant, which is not (necessarily) integrable over $\displaystyle \Omega$, and the Theorem cannot be applied as it is. But surely there is something I missed.

Thanks by advance for you help.