 Solution

For each $n$,since $S_k$ is adapted. This verifies directly that $\tau$ is a stopping time.

Set $d=b-a$. On $\{\tau >md\}$, the random walk is at some integer strictly between $a$ and $b$. If its next $d$ steps are all $+1$, it must hit $b$ within that block. These future steps are independent of ${\mathcal{F}}_{md}$ and have probability $2^{-d}$ of being all $+1$. HenceThe law of total expectation and mathematical induction give the geometric tail bound from a uniform escape probability:Therefore $\mathbb{P}(\tau <\infty )=1$. In fact this also proves $\mathbb{E}\tau \le d{2}^d<\infty$ by grouping the tail-sum formula into blocks of length $d$. No assertion of independence between successive survival events is needed.