Constant-expectation supermartingale is a martingale
= Constant-expectation supermartingale is a martingale
A discrete-time supermartingale $X$ is a martingale exactly when $\mathbb E[X_n]=\mathbb E[X_0]$ for every $n$. If expectations are constant, the nonnegative variable
$$
X_n-\mathbb E[X_{n+1}\mid\mathcal F_n]
$$
has expectation zero and therefore vanishes almost surely.