Let {$X_n$} be a uniformly integrable $(F_n)$-submartingale and $ \tau$ be the collection of all $F_n$-stopping times. Prove that {${X_T: T \in \tau}$} is uniformly integrable.
I want to use Doob's Decomposition to show that this also holds for submartingales. By Doob's Decomposition Theorem, we have that $X_n=M_n+A_n $, where $A_n $ is an increasing $F_n$-predictable process, and $M_n$ is a $(F_n)$-martingale. Now we just have to show that $M_n$ and $A_n$ are uniformly integrable. I've shown that {$M_T$} is uniformly integrable. Now I just have to show that it also holds for {$A_T$}. But I'm not sure how to proceed from here.
Please help. Thank you!
Lemma: For a collection of random variables $\{X_\alpha\}$, if $|X_\alpha| \leq Y_{\alpha}+Z$, for a non-negative collection of uniformly integrable random variables $\{Y_\alpha\}$, and a non-negative integrable random variable $Z$, then $\{X_\alpha\}$ is uniformly integrable.
Can you show this? After this, use the Doob Decomposition and the uniform integrability of $\{X_n\}$.