Let $\tau=\inf\{n:S_n=a \text{ or } S_n=-b\}$ be a stopping time and $X_n = Y_1+Y_2+...+Y_n$ a martingale (with $X_0=0$) that describes an unbalanced betting game between two players. $Y_i$ are independent random variables with $\Bbb{P}(Y_i=1)=p$ $\text{ and }$ $\Bbb{P}(Y_i=-1)=q=1-p$.
I should prove that the stopping time $\tau$ is finite almost surely.
I have found that the same question has already been asked (link: Showing a stopping time is finite), but I couldn't figure it out and solve the problem. Could someone kindly give me some suggestions? Thanks in advance.