= Stopping preserves supermartingales in discrete time
{title2=$\mathbb E[U_{(t+1)\wedge\tau}\mid\mathcal F_t]\leq U_{t\wedge\tau}$}
If $U_t$ is a <supermartingale> and $\tau$ is a <stopping time>, then $U_{t\wedge\tau}$ is a supermartingale. Adaptedness follows from the stopping-time events, and integrability follows from $|U_{t\wedge\tau}|\leq\sum_{j=0}^t|U_j|$. Its increment is $\mathbf1_{\{\tau>t\}}(U_{t+1}-U_t)$. Since $\{\tau>t\}\in\mathcal F_t$, conditioning gives a nonpositive conditional mean. No integrability assumption on the possibly unbounded stopping time is required for this stopped-process assertion at each fixed finite $t$.
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