We have , and for all , and in fact we even have where is the almost-sure (and L1) limit of using the classical theorem about uniformly integrable martingales. Thus , hence:
(using the stopping time definition) and thus finally:
In fact, for any stopping time , the stopped martingale is still uniformly integrable. You can deduce a proof from Theorem 12.5.4 (remark (i)) in these lecture notes.