= Bounded-increment optional stopping with integrable time
{title2=$\mathbb ET<\infty,\ |\Delta X_n|\le K\ \Longrightarrow\ \mathbb EX_T\le\mathbb EX_0$}
For an integrable <supermartingale> with the displayed increment bound, $|X_{T\wedge n}|\le|X_0|+KT$. Thus the stopped values are dominated by one <integrable random variable>, including $X_T$. Apply <optional stopping> at the bounded times $T\wedge n$ and then the <dominated convergence theorem>. For a <martingale> the same proof gives equality.
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