Absorption at zero of a nonnegative martingale
= Absorption at zero of a nonnegative martingale
{title2=$M_n=0\quad(n\geq T_0)$}
Once a nonnegative <martingale> hits zero, it stays zero almost surely. <Conditional expectation> on $\{M_n=0\}$ gives $\mathbb E[M_{n+1}\mathbf1_{\{M_n=0\}}]=0$; nonnegativity forces the next value to vanish there. A countable intersection makes these assertions simultaneous for all times, proving absorption at the first zero.