Consider a simple random walk in one dimension starting from the origin. Let $\epsilon>0$. How to prove that, conditioning on the event that the random walk is at the origin at time $n$, the probability that it reaches one of the sites $\pm n \epsilon$ before time $n$ goes to $0$ as $n \rightarrow \infty$?
This property should be true, as the random walk will typically explore just a region of length $\sqrt{n}$ until time $n$ and the probability that it first reaches $\pm n \epsilon$ and then it returns to the origin at time $n$ should decay much faster than the probability of just being at the origin at time $n$.