Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 211 / 2 / d / Solution

For fixed integers $s\le t$, $A_s=\{\tau =s\}$ belongs to ${\mathcal{F}}_s$ and $X_s=0$ there. The supermartingale property and nonnegativity implyTherefore $X_t=0$ almost surely on $A_s$. Taking the finite union over $s\le t$ gives $X_t=0$ on $\{\tau \le t\}$ almost surely; taking the countable intersection over all integer $t$ makes the assertion simultaneous at every time. Thus zero is absorbing for a nonnegative supermartingale in discrete time.