Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $\{X_n\}_{n\in \mathbb{N}}$ is a sequence of random variables in $L^1$.which converges to $X$ in probability and a constant $M>0 $ such that $||X_n||_1\leq M $.
Show that : $$ \mathbb {E}(|X|)\leq \liminf_{n\rightarrow\infty}(\mathbb {E}(|X_n|))\leq \sup_{n}(\mathbb {E}(|X_n|)) $$
I am thinking of applying the Fatou theorem to $\{|X_n|\}_{n\in \mathbb{N}}$, but I cannot think of a bound.