= Solution
The increments have <expected value> $p-q<0$. The <strong law of large numbers> for <independent and identically distributed random variables> with finite absolute mean gives
$$
\frac{S_n}{n}\longrightarrow p-q<0\qquad\text{almost surely}.
$$
Hence $S_n\to-\infty$. Since $q/p>1$,
$$
\boxed{Z_n=(q/p)^{S_n}\longrightarrow0\quad\text{almost surely}.}
$$
Its <expected value> nevertheless stays equal to one. Thus this <martingale> is not <uniformly integrable>, and it does not converge to zero in $L^1$. This distinction will be avoided in part (ii) by stopping at an upper level, which makes the stopped process bounded.
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