Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 201 / 2 / a / Solution

For each deterministic $n$, use the nonnegativity and the martingale property to obtainA 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 givesZero 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.