= Solution
For a <martingale> and a <stopping time> $T\leq N$, expand the stopped value using its <martingale differences>:
$$
M_T-M_0=\sum_{k=1}^N\mathbf1_{\{T\geq k\}}(M_k-M_{k-1}).
$$
The event $\{T\geq k\}=\{T>k-1\}$ belongs to the <filtration> $\mathcal F_{k-1}$. Each summand is integrable and its <conditional expectation> given $\mathcal F_{k-1}$ is zero. Taking <expectations> of this finite sum therefore proves \b[$\mathbb E M_T=\mathbb E M_0$ for every bounded stopping time]. This proves the forward direction of the <bounded-stopping-time characterization of a martingale> without assuming an <optional stopping theorem>.
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