According to Wikipedia,
A class $\mathcal{C}$ of random variables is $\textbf{uniformly integrable}$ if given $\epsilon > 0$, there exists $K \in [0. \infty)$ such that $\textbf{E}(|X|I_{|X| \geq K}) \leq \epsilon$ for all $X \in \mathcal{C}$.
It says this is equivalent to saying that $\lim_{K \to \infty} \sup_{x \in \mathcal{C}}\textbf{E}(|X|I_{|X| \geq K})=0$. How to show this?
My attempt: It is clear that for $K < M$, $|X|I_{|X| \geq M} \leq |X|I_{|X| \geq K}$. Therefore, for this $\epsilon$, $\textbf{E}(|X|I_{|X| \geq M}) \leq \epsilon$ for all $M \geq K$. Hence, $\lim_{K \to \infty} \sup_{x \in \mathcal{C}}\textbf{E}(|X|I_{|X| \geq K})=0$. Is my reasoning correct? Also, is there an easier proof?
Your reasoning is correct but shows only one direction. For the other, assume that $\lim_{R \to \infty} \sup_{x \in \mathcal{C}}\textbf{E}(|X|I_{|X| \geq R})=0$. For a fixed $\varepsilon>0$, choose $K $ such that for $R\geq K$, $\sup_{x \in \mathcal{C}}\textbf{E}(|X|I_{|X| \geq R})\leq\varepsilon$.