= Solution
The first-contact time $\tau_*\le T$ is a <stopping time>: the event $\{\tau_*\le t\}$ is the finite union of events $\{U_s=Y_s\}$ for $s\le t$. It is finite because $U_T=Y_T$. On $\{\tau_*>t\}$ we have $U_t>Y_t$, so the maximum defining the <Snell envelope> selects continuation and
$$
U_t=\mathbb E(U_{t+1}\mid\mathcal F_t).
$$
The stopped process $U_{t\wedge\tau_*}$ is therefore a <martingale>: before stopping its conditional increment is zero and after stopping its increment vanishes. Applying the stopped-sum argument with zero conditional increments gives
$$
\boxed{U_0=\mathbb E U_{\tau_*}=\mathbb E Y_{\tau_*}.}
$$
The upper bound from part (a) now proves that first contact is an <optimal stopping time>.
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