Bounded-increment martingale convergence-or-oscillation dichotomy ($X_n\to X_{\infty }\in \mathbb{R}\text{or}\mathrm{liminf}{X}_n=-\infty ,\mathrm{limsup}{X}_n=+\infty$)

A discrete-time martingale with a common deterministic bound on all increments either converges finitely or has $\mathrm{liminf}{X}_n=-\infty$ and $\mathrm{limsup}{X}_n=+\infty$, almost surely. Stopping on first crossing below $-A$ gives a lower bound $-A-M$ on the overshoot. Adding $A+M+X_0^{-}$ produces a nonnegative martingale, so the Martingale convergence theorem forces finite convergence on the event of never crossing. Apply the same argument to $-X$.