Solution
= Solution
Under the standard <local martingale> definition, $X_0$ is integrable. For a nonnegative $X$, the <Fatou lemma> and a <localizing sequence> give
$$
\mathbb EX_t\leq\liminf_n\mathbb EX_{t\wedge\tau_n}=\mathbb EX_0<\infty.
$$
Hence every $X_t$ is integrable, and part (a) applies. Therefore \b[a nonnegative discrete-time <local martingale> is a true <martingale>.] This is stronger than the continuous-time <supermartingale> conclusion.