= Solution
For each deterministic $n$, use the nonnegativity and the <martingale> property to obtain
$$
\mathbb E\left[M_{n+1}\mathbf1_{\{M_n=0\}}\right]
=\mathbb E\left[\mathbf1_{\{M_n=0\}}\mathbb E[M_{n+1}\mid\mathcal F_n]\right]=0.
$$
A nonnegative random variable with zero <expectation> is zero almost surely. Hence $M_{n+1}=0$ almost surely on $\{M_n=0\}$. Intersecting these probability-one statements over all $n$ makes them hold simultaneously, and induction gives
$$
\boxed{M_n=0\text{ for every }n\geq T,\quad\text{almost surely}.}
$$
\b[Zero is absorbing for a nonnegative martingale.] This is the <absorption at zero of a nonnegative martingale>. On $\{T=\infty\}$ the assertion has no finite-time requirement. No unbounded-time use of the <optional stopping theorem> is needed.
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