= Solution
Let $\tau_n\uparrow\infty$ be a <localizing sequence> for the discrete-time <local martingale> $X$. For $t\geq1$, the increment of the stopped <martingale> is
$$
X_{t\wedge\tau_n}-X_{(t-1)\wedge\tau_n}=\mathbf1_{\{\tau_n\geq t\}}(X_t-X_{t-1}),
$$
and $\{\tau_n\geq t\}\in\mathcal F_{t-1}$. Thus, for any $A\in\mathcal F_{t-1}$,
$$
\mathbb E[\mathbf1_A\mathbf1_{\{\tau_n\geq t\}}(X_t-X_{t-1})]=0.
$$
The assumed integrability gives the dominating <random variable> $|X_t|+|X_{t-1}|$. The <dominated convergence theorem> now yields $\mathbb E[\mathbf1_A(X_t-X_{t-1})]=0$, or $\mathbb E[X_t\mid\mathcal F_{t-1}]=X_{t-1}$. Iterating the <tower property of conditional expectation> proves \b[an integrable discrete-time <local martingale> is a true <martingale>.] The proof uses the discrete-time increment indicator; the conclusion does not extend to continuous time.
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