Let $S\subsetneq\mathbb{Z}$ be an infinite set. Does there always exist a prime $p$ such that the closure of $S$ in the $p$-adic integers, $\mathbb{Z}_p$, contains a rational integer $n\notin S$?
Or, in elementary language, does there always exist a prime $p$ and $n\in \mathbb{Z}\setminus S$ such that for all $k$ there is an $s\in S$ such that $p^k\mid n-s$.