Let $X$ be a symmetric nearest neighbor random walk in $\mathbb{Z}$ with $X_0 = 0$ and let $a,b \in \mathbb{N}$. Set $\mathcal{F}_n = \sigma(X_0,X_1,...,X_n)$. Show that $\tau = \inf \{n \in \mathbb{N}: X_n \in \{a, -b\}\}$ is a stopping time.
My argument looks like this:
$\{\tau \leq n\} = \bigcup_{k=0}^n\{X_k \in \{a,-b\}\} \in \mathcal{F}_n$ since $\{X_k \in \{a,-b\}\} \in \mathcal{F}_k \subset \mathcal{F}_n$.
But I'm not sure if there is something missing.