Hello!

Could you please help me with this question:

Prove the following (known as Pratt's Lemma):

Let be sequences of integrable functions. If

1) as

2)

3) and as with and both finite,

then

and is finite.

I was thinking about mimicking the proof of the Dominated Convergence Theorem (noting that and by Fatou's Lemma, concluding that ) but I'm not quite sure where to go from there... Any hints/help is appreciated