I have the following situation. Let $(X_i)_{i\geq1}$ be a sequence of iid random variables in $\mathbb{Z}$ and consider the random walk $S_n=\sum_{i=1}^n{X_i}$, $S_0=x$. Let $y>x$ and consider the set $A=(0,y)$ I want to show that the the first leaving time $T_A$ of the set $A$ is finite a.s.. But it seems to be not that trivial.
Intuitively whenever I have more than $y$ consecutive rv taking values bigger or equal than $1$, then I leave the set, and so I could reduce the problem by checking that this event has measure 1. But it does not seems very trivial.
There is a positive probability that after $y$ steps, we will get out of the interval (by going to the left each time, for example). Call this probability $p$. Then $T_A$ is bounded by a geometric random variable with parameter $p$. Thus it is finite a.s.