= Uniform integrability of a stopped uniformly integrable martingale
Stopping a <uniformly integrable> discrete-time <martingale> at any <stopping time> preserves <uniform integrability>, including when the <stopping time> can be infinite. For $C=\sup_n\mathbb E|X_n|$, the <optional stopping theorem> and the <Markov inequality> give
$$
\mathbb E[|X_{n\wedge T}|\mathbf1_{\{|X_{n\wedge T}|>K\}}]\leq\sup_j\mathbb E[|X_j|\mathbf1_{\{|X_j|>R\}}]+RC/K.
$$
Choose $R$ first, then $K$, to make this uniformly small.
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