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.
You don't require any bound on $E|X_n|$ for this. You can prove it using Fatou's Lemma.
Suppose $E|X| >\lim \inf E|X_n|$. Then we can find a real number $c$ and integers $n_1<n_2<...$ such that $E|X| >c >E|X_{n_k}|$ for all $k$. Since $X_{n_k} \to X$ in probability there is a subsequence, call it $Y_j$ which converges to $X$ almost everywhere. Bu then $E|X| >c >E|Y_j|$ for all $j$ and $Y_j$ converges to $X$ almost everywhere. This is a contradiction to Fatou's Lemma.
Choose a subsequence $(n_k)_{j\geq 1}$ such that
$$\liminf_{n\to\infty} \mathbb{E}[|X_n|] = \lim_{k\to\infty} \mathbb{E}[|X_{n_k}|]$$
holds. Since $X_{n_k} \to X$ in probability, we may find a further subsequence $(n'_j)_{j\geq 1} \subseteq (n_k)_{k\geq 1}$ such that $X_{n'_j} \to X$ almost surely as $j\to\infty$. Then by the Fatou's lemma,
$$ \liminf_{n\to\infty} \mathbb{E}[|X_n|] = \lim_{j\to\infty} \mathbb{E}[|X_{n'_j}|] \geq \mathbb{E}\Bigl[ \liminf_{j\to\infty} |X_{n'_j}| \Bigr] = \mathbb{E}[|X|]. $$
