 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 sumThat sum is integrable under the hypothesis. Since $X^{{\tau }_j}$ is a martingale,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 $X$ 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.