How to prove $EX_n\uparrow EX$?

Question

How to prove $EX_n\uparrow EX$? The question is as follows.

If $EX_1^- < \infty$ and $X_n \uparrow X$, then $EX_n \uparrow EX$.

Maybe using monotone convergence theorem, but I really have no clue how to apply.

Answer 1

It follows from the monotonicity that

$$\mathbb{E}X_n \leq \mathbb{E}X.$$

On the other hand, $$Y_n := X_n+X_1^-$$ defines a sequence of non-negative random variables and by Fatous lemma

$$\mathbb{E}\left( \lim_{n \to \infty} Y_n \right) \leq \liminf_{n \to \infty} \mathbb{E}(Y_n).$$

Conclude from this inequality that

$$\mathbb{E}X \leq \liminf_{n \to \infty} \mathbb{E}X_n.$$