Why is this true of a submartingale?

Question

Suppose $(X_n, \mathbb F_n)$ is a submartingale. A proof that I am currently reading seems to use that $$EX_n \geq E X_0$$ Is that true in general? If so, how does it follow from the submartingale property?

Answer 1

If $X_n$ is a submartingale, then $\mathbb{E}[X_n\mid \mathbb{F}_{n-1}] \geq X_{n-1}$. Taking the expectation of both sides, we have \begin{align*} \mathbb{E}[\mathbb{E}[X_n \mid \mathbb{F}_{n-1}]] &\geq \mathbb{E}[X_{n-1}]\\ \mathbb{E}[X_n] &\geq \mathbb{E}[X_{n-1}]. \end{align*} This holds for all $n$, so stringing these inequalities together gives $$\mathbb{E}[X_n] \geq\mathbb{E}[X_{n-1}] \geq \cdots \geq \mathbb{E}[X_1] \geq \mathbb{E}[X_0].$$