An adapted RCLL process $(X_t)_{t\leq T}$ is said to be of class $(D)$ is the family $$ \{ X_\tau \mid \tau\leq T \text{ is a stopping time}\}$$ is uniformly integrable.
I've heard a lemma that every supermartingale $(X_t)_{t\leq T}$ is locally of class $(D)$ but could not find a proof. Can anyone show how this is done?
Define stopping time $\tau_n=\inf\{t\in [0,T]:|X_t|>n\}$ then for any stopping time $\sigma$ you have that $$|X_{\tau_n\wedge\sigma}|\leq n+|X_{\tau_n}|$$ so it suffices to argue that $|X_{\tau_n}|\in L^1$ but notice that since $X$ is a super martingale, $(X)^{+}+(X)^-=|X|$ are sub-martingales so $$|X_{\tau_n}|\leq\mathbb{E}[|X_T|\vert \mathcal{F}_{\tau_n}]\in L^1$$ thus concluding that every super-martingale is locally of class (D).