= Solution
The time $T_R$ is a <stopping time>, because $\{T_R\leq n\}=\bigcup_{k\leq n}\{M_k\geq R\}\in\mathcal F_n$. Apply the <optional stopping theorem> to the bounded time $T_R\wedge n$:
$$
1=\mathbb E M_{T_R\wedge n}
\geq R\,\mathbb P(T_R\leq n).
$$
On $\{T_R\leq n\}$ the stopped value is at least $R$, and elsewhere it is nonnegative. Letting $n\to\infty$ gives the <maximal bound for a nonnegative martingale>:
$$
\boxed{\mathbb P(T_R<\infty)\leq\frac1R.}
$$
\b[Stop at a bounded time first, then pass to the increasing event.] This avoids assuming <uniform integrability> of the original process. For $0<R\leq1$, $T_R=0$ and the bound simply says $1\leq1/R$.
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