Let $\{X_n\,:\,n=1,2,...\}$ be a sequence of random variables. Define \begin{align*} F_n=\sigma(X_1,X_2,...,X_n). \end{align*} Suppose that for any bounded stopping times $\sigma\le\tau$, we have \begin{align*} EX_\sigma\le EX_\tau,\quad\text{a.s.} \end{align*} Prove that $(X_n, F_n)$ is a submartingale.
Unfortunately I don't have any preliminary work for this one, I know multiple theorems about stopping times for martingales but they $\textbf{assume}$ that the $X_n$ already make up a martingale. So, I am at a loss for how to even begin this one, any help would be greatly appreciated.