Let $\left(X_{n}\right)_{n \in \mathbb{N}_{0}}$ be a submartingale or a supermartingale. Use Theorem $11.2$ and Doob's decomposition to show that, for all $n \in \mathbb{N}$ and $\lambda>0$,
$$
\lambda \mathbf{P}\left[|X|_{n}^{*} \geq \lambda\right] \leq 12 \mathbf{E}\left[\left|X_{0}\right|\right]+9 \mathbf{E}\left[\left|X_{n}\right|\right].
$$