It looks like Doob's inequality, but goes the other direction.

Question

$(X_n)$ is a nonnegative martingale with $X_0=1$ and $X_n$ converges to $0$ a.s. Suppose that $X_n$ only has finitely many possible values for each $n$. Take $S_n=\max\{x:P(X_{n+1}=x\mid F_n)>0\}$. Prove that $P(\sup_{n\geq0}S_n\geq\lambda)\geq1/\lambda,\quad\lambda>1$.