Show that every bounded discrete subharmonic function is constant.

Question

I have to show that every bounded discrete subharmonic function is constant, using the fact that when $f: \mathbb{Z} \rightarrow \mathbb{R}$ is discrete subharmonic, then $f(S_n)$ with $S_n$ a random walk, is a submartingale. I also must use the recurrence of $S_n$, i.e. $S_n$ visits every point $x \in \mathbb{Z}$ infinitely many times with probability one.