Let $(X_n)_{n\in\mathbb N_0}$ be a submartingale or a supermartingale. Show that, for all $n\in\mathbb N$ and $\lambda>0$,
$$\lambda P[|X|^*_n\ge \lambda ]\le 12 E[|X_0|]+9E[|X_n|].$$
This problem comes from Klenke's probability textbook (11.1.1), and follows the section on Doob's inequalities and Doob's decomposition. We define
$$X^*_n = \sup\{X_k : k\le n\}.$$
My thoughts: For simplicity, consider just the case where $X_n$ is a submartingale. Then we can write $X_n=M_n+A_n$, where $M_n$ is a martingale and $A_n$ is an increasing predictable process (so a positive submartingale). Then
$$P[|X|^*_n\ge \lambda ] \le P[|A|^*_n+|M|_n^*\ge \lambda ] \le P[|M|_n^*\ge a\lambda ]+ P[|A|^*_n\ge (1-a)\lambda ],$$
where $a$ will be chosen later. Combining this with the inequality
$$\lambda P[|Y|^*_n\ge \lambda ]\le E[|Y_n|]$$
(which applies with $Y_n$ is a martingale or positive submartingale) gives a bound in terms of $A_n$ and $M_n$, which is not quite what we want. We need a bound in terms of $|X_0|$ and $|X_n|$. I am unable to proceed and would appreciate any help.
Note that it suffices to consider the case when $X$ is a submartingale since if $X$ is a supermartingale then $-X$ is a submartingale.
To finish your proof, use the following bounds: $$E[|A_n|] = E[A_n] = E[X_n - M_n] = E[X_n] - E[M_n] = E[X_n] - E[M_0] = E[X_n] - E[X_0] \leq E[|X_n|] + E[|X_0|]$$ and $$E[|M_n|] = E[|X_n-A_n|] \leq E[|X_n|] + E[|A_n|] \leq 2 E[|X_n|] + E[|X_0|].$$