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?
Note that for any $K, n\in\mathbb{N}$ with $K<n$ we have that $$\begin{align*} \mathbb{E}[|X_n|1_{X_n\in[-n, n]}1_{|X_n|>K}] &= \mathbb{E}[|X_n|1_{n \geq |X_n|>K}]\\ &=\int_{n\geq |X_n|>K} |X_n|d\mathbb{P}\\ &= \sum_{j=K+1}^n \int_{j\geq |X_n|>j-1} |X_n|d\mathbb{P}\\ &\leq \sum_{j=K+1}^n M_j+1/j^2\\ &\leq \left(\sum_{j=K+1}^\infty M_j\right)+ \left(\sum_{j=K+1}^\infty 1/j^2\right) \end{align*} $$
Note that the final bound does not depend on $n$, so $\sup_n \mathbb{E}[|X_n|1_{X_n\in[-n, n]}1_{|X_n|>K}]\leq \left(\sum_{j=K+1}^\infty M_j\right)+ \left(\sum_{j=K+1}^\infty 1/j^2\right)\overset{K\to\infty}{\longrightarrow} 0$. Hence, we have uniform integrability.
