= Solution
Choose a <localizing sequence> of discrete-time <stopping times> $\tau_j\uparrow\infty$ for $X$. For fixed integer $t$, every stopped value is bounded in absolute value by the finite sum
$$
|X_{t\wedge\tau_j}|\leq\sum_{u=0}^t|X_u|.
$$
That sum is <integrable> under the hypothesis. Since $X^{\tau_j}$ is a <martingale>,
$$
\mathbb E[X_{t\wedge\tau_j}\mid\mathcal F_{t-1}]
=X_{(t-1)\wedge\tau_j}.
$$
The <dominated convergence theorem>, including its conditional version, now removes the stopping. Thus $\mathbb E[X_t\mid\mathcal F_{t-1}]=X_{t-1}$. The process is <adapted> and <integrable> by hypothesis, so \b[$X$ is a true discrete-time <martingale>]. This is the <integrable discrete-time local martingale is a martingale> criterion. The finite sum dominating stopped values is the crucial discrete-time feature.
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