Say there is a random walk $\{S_n\}$ with $S_0=0$ and $0<p=P(S_1=1)<\frac{1}{2}$. We know such a random walk would go to $-\infty$ eventually. Define the stopping time $T=\inf\{n: S_n=-\infty\}$, how can we argue that the stopping time is finite a.s., i.e., $P(T<\infty)=1$?
I know we definitely will use $0<p=P(S_1=1)<\frac{1}{2}$, and I know for such $p$, the returning time is not finite, which is transient. How can I apply this to show the stopping time $T$ is finite a.s.? Or is there another approach to this?