 Solution

For $q>p$, let $M={\sup }_{n\ge 0}{S}_n$, 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) giveThus $\boxed{\mathbb{E}\underset{n\ge 0}{\sup }{S}_n\le p/(q-p)}$, and the supremum is finite almost surely. The sum identity follows by writing $M={\sum }_{k\ge 1}{1}_{\{M\ge k\}}$ and applying monotone convergence theorem. If the printed supremum is read literally over $n\ge 1$, 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.