Will a nonnegative supermartingale converge to 0?

Question

I want to know weather a non negative supermartingale converges to $0$.

I have a hunch that it shall be so, but could not prove or disprove it. Is this correct? And if so, is there a way to prove it?

Answer 1

No, this does not hold. If $(X_n)$ is a non-negative supermartingale converging to $0$, then $Y_n := X_n+1$ is still a supermartingale and $\lim (Y_n) = \lim(X_n)+1 = 1$.