Let $X_1,...,X_t$ be an i.i.d. sequence of random variables with support $\{a,-b\}$, where $a,b>0$, and measure $P(a)=p_1$, $P(-b)=p_2$. Assume $p_1a-p_2b<0$, so that $E[X_t]<0$.
Let $S_t=\sum_{i=1}^t X_i$. It is straightforward to see that $S_t \rightarrow -\infty$.
Let $\tau$ be the stopping time corresponding to the first time that $S_t$ exceeds $M$ for some $M \in (-b,0]$, $$\tau=\inf\{t \geq 1 \mid S_t \geq M\}$$ I am trying to figure out how to show that $P(\tau=\infty)>0$.