Maximal bound for a nonnegative martingale ($\mathbb{P}(T_R<\infty )\le \mathbb{E}{M}_0/R$)

For a nonnegative martingale, $T_R=\inf \{n:M_n\ge R\}$ satisfies $\mathbb{P}(T_R<\infty )\le \mathbb{E}{M}_0/R$. Stop at $T_R\wedge n$ and use nonnegativity before passing to the increasing event. A limit of thresholds from below also gives the bound for $\{{\sup }_n{M}_n\ge R\}$ even when the supremum is not attained.