Integrable discrete-time local martingale is a martingale

An integrable discrete-time local martingale is a true martingale. Its stopped increments have the form $1_{\{\tau \ge t\}}(X_t-X_{t-1})$, and the dominated convergence theorem removes a localizing sequence from the conditional increment identity.

The Fatou lemma makes every time value of a nonnegative discrete-time local martingale integrable, assuming the usual integrable initial value. The integrable discrete-time result then makes it a true martingale.