I have read lots of proofs for the Lebesgue Dominated Convergence Theorem and a lot of them involves Fatou's Lemma. My question is can we directly prove LDCT without using Fatou's Lemma? We study measure theory but we did not cover Fatou's Lemma yet and I dont have any idea how to start the proof.
My first guess is we use Monotone Convergence Theorem but the monotonicity of the sequence is not part of the assumption. I don't know any other theorem that we discussed that involves limits hence I do not know where to start.
Any lead, hint or help will be much appreciated. Thank you!