The supermartingale convergence theorem says that if $X_n \geq 0$ Is a supermartingale, then $X_n \to X$ a.s for some $X$. Further, $EX \leq EX_0$.
My question is whether the following is a valid application of this theorem:
Suppose $X_n$ is a non-negative martingale. Then it is also a supermartingale and hence converges a.s to some $X$ a.s
Since $EX \leq EX_0 <\infty$ it must be that $X <\infty a.s$. Hence any non-negative martingale almost surely converges to a finite limit.