For a random walk $S_n$, does a uniform bound $P(S_n \in A) < \delta$ give a bound on the time spent by the random walk in $A$?

Question

Title pretty much says all. Let $S_n$ be a symmetric random walk on the integers, and $A$ some set of integers. Assume that we have a bound $P(S_n \in A) < \delta$ for all $n$. Can we say something about $P\left(\frac{\#\{n = 0, \cdots, N \mid S_n \in A\}}{N} > \epsilon\right)$