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

The time $T_R$ is a stopping time, because $\{T_R\le n\}={\bigcup }_{k\le n}\{M_k\ge R\}\in {\mathcal{F}}_n$. Apply the optional stopping theorem to the bounded time $T_R\wedge n$:On $\{T_R\le n\}$ the stopped value is at least $R$, and elsewhere it is nonnegative. Letting $n\to \infty$ gives the maximal bound for a nonnegative martingale:Stop at a bounded time first, then pass to the increasing event. This avoids assuming uniform integrability of the original process. For $0<R\le 1$, $T_R=0$ and the bound simply says $1\le 1/R$.