Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability.

Suppose $\{X_n\}$ is a sequence of random variables satisfying : $$ \sup_{n}{\mathbb{E}(|X_n|)} <\infty $$ Suppose that $$ \dfrac{M_j}{2}<\int_{j-1<|X_{n}|\leq j}{|X_{n}(t)|d\mathbb{P}(t)}\leq M_j+\dfrac{1}{j^2} \qquad\forall n\geq 1 \text{ and }1\leq j\leq n^2 $$ with $M_j>0$ such that $\sum_{j=1}^{\infty}{M_j}<\infty$.

Show that: $$ \{X_n\mathbb{1}_{X_n\in[-n,n]}\}\text{ is uniformly integrable} $$

Any ideas, please?

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space.

Suppose $\{X_n\}$ is a sequence of random variables satisfying : $$ \sup_{n}{\mathbb{E}(|X_n|)} <\infty $$ Suppose that $$ \dfrac{M_j}{2}<\int_{j-1<|X_{n}|\leq j}{|X_{n}(t)|d\mathbb{P}(t)}\leq M_j+\dfrac{1}{j^2} \qquad\forall n\geq 1 \text{ and }1\leq j\leq n^2 $$ with $M_j>0$ such that $\sum_{j=1}^{\infty}{M_j}<\infty$.

Show that: $$ \{X_n\mathbb{1}_{X_n\in[-n,n]}\}\text{ is uniformly integrable} $$

Any ideas, please?