 Absorption at zero of a nonnegative martingale

Once a nonnegative martingale hits zero, it stays zero almost surely. Conditional expectation on $\{M_n=0\}$ gives $\mathbb{E}[M_{n+1}{1}_{\{M_n=0\}}]=0$; nonnegativity forces the next value to vanish there. A countable intersection makes these assertions simultaneous for all times, proving absorption at the first zero.