
Limit set question
Suppose you have an infinite tower of subsets of the integers: $\displaystyle A_1 \subsetneq A_2 \subsetneq \cdots \subset \mathbb{Z}$ such that for any $\displaystyle x \in \mathbb{Z}$, there exists $\displaystyle k \in A_n$ such that $\displaystyle x \equiv k \pmod{2^n}$. Can I determine whether or not $\displaystyle x \in \bigcup_{n\ge 1}A_n$?

Re: Limit set question
I figured out the answer is no. For example, if $\displaystyle A_n = [2n] = \{1,\ldots, 2n\}$, then this satisfies the claim that for any $\displaystyle x \in \mathbb{Z}$, there exists $\displaystyle k \in A_n$ with $\displaystyle x \equiv k \pmod {2^n}$. However, $\displaystyle \bigcup_{n\ge 1} A_n = \mathbb{N}$, so it would not contain any nonpositive integers.