Application of Doob's Optional Stopping Time Theorem on new stopping time

Question

Consider a random walk on a line starting at 0. On each step the probability of moving in either direction (right or left) is 1/2. There are two particular points on the line -a, and b. If $\tau$ is the hitting time of A, then $\tau$ is not bounded, and we cannot apply Doob's Theorem. However, we can apply Doob'ts Theorem to $\tau'$ = $\min(\tau, n)$. I don't understand why $\min(\tau, n)$ is necessarily bounded.. can't the 'walker' stay between -a and b to $\infty$? so then n is $\infty$?