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.