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For $A\in {\mathcal{F}}_n$, define the bounded stopping time $T=n+1$ on $A$ and $T=n$ on $A^c$. Before time $n$ its stopping events are empty, at time $n$ the event $\{T\le n\}=A^c$ belongs to ${\mathcal{F}}_n$, and subsequently it has stopped everywhere. Subtract the assumed mean identity for this $T$ from that for the constant stopping time $n$ to obtainSince $M_n$ is integrable and ${\mathcal{F}}_n$-measurable, this is precisely the defining property of conditional expectation giving $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$. Thus the adapted process is a martingale, completing the bounded-stopping-time characterization of a martingale.

For a martingale and a stopping time $T\le N$, expand the stopped value using its martingale differences:The event $\{T\ge k\}=\{T>k-1\}$ 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 $\mathbb{E}{M}_T=\mathbb{E}{M}_0$ 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.