Stopping preserves supermartingales in discrete time ($\mathbb{E}[U_{(t+1)\wedge \tau }\mid {\mathcal{F}}_t]\le U_{t\wedge \tau }$)

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 $|U_{t\wedge \tau }|\le {\sum }_{j=0}^t|U_j|$. Its increment is $1_{\{\tau >t\}}(U_{t+1}-U_t)$. 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 $t$.