Given a positive integer $n\in\mathbb{N}$ we define the *zebra crossing* associated with $n$ to be the set $$Z_n = \{[2kn, 2kn+(n-1)] \cap \mathbb{N}: k\in\mathbb{N}\}.$$

Is there an infinite set $A\subseteq\mathbb{N}$ such that $A\cap Z_n$ is finite for all positive integers $n$?