= Solution
For $q>p$, let $M=\sup_{n\geq0}S_n$, including $S_0=0$. It is a nonnegative integer-valued <random variable>, possibly infinite before the following estimate. The <tail-sum formula for expectation> and part (i) give
$$
\mathbb EM=\sum_{k\geq1}\mathbb P(M\geq k)
\leq\sum_{k\geq1}(p/q)^k
=\frac{p/q}{1-p/q}=\frac p{q-p}.
$$
Thus $\boxed{\mathbb E\sup_{n\geq0}S_n\leq p/(q-p)}$, and the supremum is finite <almost surely>. The sum identity follows by writing $M=\sum_{k\geq1}\mathbf1_{\{M\geq k\}}$ and applying <monotone convergence theorem>. If the printed supremum is read literally over $n\geq1$, its positive part has exactly the same positive-level tails and its negative part is bounded by one. The same upper bound on its <expected value> remains valid.
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