Lebesgue integral & Mathematical expectation

Let a sequence of random variables on a probability space such that for some constant . Assume that almost surely as . How do you prove that is finite and ?

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 , and the Theorem cannot be applied as it is. But surely there is something I missed.

Thanks by advance for you help.