= Solution
Let $U=U_0+M-A$ be its <Doob decomposition in discrete time> and choose the martingale
$$
X_t^*=-M_t.
$$
Since $Z_t\leq U_t$ and $A_t\geq0$,
$$
Z_t+X_t^*
\leq U_t-M_t
=U_0-A_t
\leq U_0.
$$
For the first optimal stopping time $\tau^*$, the <complementarity for the Snell envelope compensator> implies $A_{\tau^*}=0$: all compensator increments before $\tau^*$ vanish. Since $Z_{\tau^*}=U_{\tau^*}$,
$$
Z_{\tau^*}+X_{\tau^*}^*=U_0.
$$
Therefore
$$
\max_{0\leq t\leq T}(Z_t+X_t^*)=U_0
$$
pathwise, and taking expectations proves the asserted equality.
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