Let $S_{n}$ be a symmetric random walk starting at $0$. That is $S_{n}= \xi_{1} + \cdots \xi_{n}$ where the $\xi_{i}$ are i.i.d. and $P(\xi_{i}=1)=1/2$ and $P(\xi_{i}=-1)=1/2$. I want to be able to show that the following stopping times:
$$\tau := \inf \{n \in \mathbb{N}: S_{n} \notin (a,b) \}$$
are finite almost surely using the martingale convergence theorem and either the Kolmogorov zero-one law or the Hewitt Savage zero-one law.
The idea is to consider the stopped processes:
$S_{n \wedge \tau}$ which are bounded martingales and hence converge to $Y$ almost surely. In particular, we have that
\begin{align} \textbf{1}_{\tau = \infty}S_{n \wedge \tau} \to \textbf{1}_{\tau = \infty}Y \\ \end{align} and therefore: \begin{align} \textbf{1}_{\tau = \infty}S_{n} \to \textbf{1}_{\tau = \infty}Y \\ \\ \end{align} I feel like I should be able to conclude that $\tau < \infty$ almost surely using one of the two zero-one laws above.