= Bounded-increment martingale convergence-or-oscillation dichotomy
{title2=$X_n\to X_\infty\in\mathbb R\quad\text{or}\quad\liminf X_n=-\infty,\ \limsup X_n=+\infty$}
A discrete-time <martingale> with a common deterministic bound on all increments either converges finitely or has $\liminf X_n=-\infty$ and $\limsup X_n=+\infty$, <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 $-X$.
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