Fatou's Lemma: How to apply it for $E[|X|] \leq \liminf_n E[|X_n|]$

Question

Let $X_n$ for $n \geq 0$ be a sequence of random variables with $\sup_n E[X_n^+] < \infty$ and $X_n \overset{P-a.s.}{\rightarrow} X$.

We have shown that $$E[X_0] \leq E[X_n ] =E[X_n^+]-E[X_n^-]< \infty$$

and that

$$E[X_n^-] \leq E[X_n^+] - E[X_0] < \infty.$$

Now we conclude

1. $\sup_n E[|X_n|] < \infty$
2. $E[|X|] \leq \liminf_n E[|X_n|] < \infty$ using Fatou's Lemma

Can you help me providing some more Details for These two conclusions?

1. $\sup_n E[|X_n|] = \sup_n E[\max(X_n^+, X_n^-)]< \infty$, since the above inequalities are Independent of $n$, so $E[X_n^+]$ and $E[X_n^-]$ are uniformly bounded?
2. Here I have no idea what the supremum has to do with $\liminf_n$, so we would Need to write $E[|X|] \leq \dotsc \leq E[\liminf_n |X_n|] \leq \liminf_n E[|X_n|] < \infty$.

You see, I Need some Details here how to convert $|X|$ to $X_n^+$, $X_n^-$.. Thanks a lot!

Answer 1

Since $\mathbb E[X_0]\leqslant \mathbb E[X_n]$ for each $n$, we have $$\sup_n\mathbb E[X_n^-]\leqslant \sup_n\mathbb E[X_n^+]-\mathbb E[X_0]$$ and the right hand side is finite by assumption.

For 2., we apply Fatou's lemma to the non-negative sequence $\left(\left|X_n\right|\right)_{n\geqslant 1}$ (which converges almost everywhere to $\left|X\right|$).