Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 5 / b / Solution

Let ${\tau }_n$ localize the nonnegative local martingale $M$. For $s\le t$,Conditional Fatou lemma and nonnegativity giveThus $M$ is a supermartingale, recovering the general fact about a nonnegative local martingale.