= Solution
Let $r=q/p>1$ and $H_k$ be the first <hitting time> of level $k$. Nearest-neighbour steps imply $S_{H_k}=k$ whenever $H_k<\infty$: the walk cannot overshoot its first upward crossing. On this event the stopped <martingale> eventually has the constant value $r^k$. On $\{H_k=\infty\}$, it is $Z_n$ and tends to zero by part (i). Therefore
$$
\boxed{Z_{n\wedge H_k}\longrightarrow r^k\mathbf1_{\{H_k<\infty\}}
=e^{\lambda_k}\mathbf1_{\{H_k<\infty\}},\qquad\lambda_k=k\log(q/p).}
$$
For every $n$, $0\leq Z_{n\wedge H_k}\leq r^k$. Before the hit the walk is below $k$, and at the hit it equals $k$. The <optional stopping theorem> for the bounded <stopping time> $n\wedge H_k$ gives $\mathbb EZ_{n\wedge H_k}=\mathbb EZ_0=1$. The <dominated convergence theorem> now yields $1=r^k\mathbb P(H_k<\infty)$. Consequently
$$
\boxed{\mathbb P(H_k<\infty)=(p/q)^k.}
$$
The uniform bound on the stopped variables justifies the limit; applying optional stopping directly at the possibly infinite $H_k$ without this argument would not be valid.
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