# Doob's Decomposition Theorem and Uniform Integrability

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.

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 $$\{X_n\}$$.