Why is every bounded local martingale a true martingale?

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A cadlag process is a local martingale $\iff$ there exists a sequence of stopping times $\tau_n\to\infty (n\to\infty)$ a.s. and the stopped process $M^{\tau_n}$ is a true martingale for all $n\in\mathbb{N}$. Why is every bounded local martingale a true martingale?