$S_{n}$ is a submartingale, $\sup_{n\geq0}E[|S_{n}|] < \infty \Leftarrow \sup_{n\geq0}E[S_{n}^{+}] <\infty$

Question

$\sup_{n\geq0}E[|S_{n}|] < \infty \Leftarrow \sup_{n\geq0}E[S_{n}^{+}] <\infty$ is apparently true. The converse is easy to prove and thus it is an iff statement. I have tried a few different things (see comments) but a point in the right direction would certainly be much appreciated.

Answer 1

Based on the kind of things you've been trying, this is a lot simpler than you think.

Since $S_n$ is a submartingale, $$\mathbb{E}[S_n] = \mathbb{E}[S_n^+ - S_n^-] \geq \mathbb{E}[S_0].$$ Rearranging gives us that $\mathbb{E}[S_n^-] \leq \mathbb{E}[S_n^+] - \mathbb{E}[S_0]$ which will lead to the desired bound by writing $|S_n| = S_n^+ + S_n^-$.