A sequence of subsets of $\Bbb Z$ not containing nontrivial subgroups

Question

Is there a sequence $(A_n)$ of subsets of $\Bbb Z$ such that always $\{a-b\mid a,b\in A_{n+1}\}$ is a proper subset of $A_n$ and no $A_n$ contains an infinite subgroup of $(\Bbb Z,+)$?

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^{(Ed.: this is a self-answered question)}

Answer 1

Yes. No proof is requested. But to show the correctness of the answer: Embed $\Bbb Z$ in the circle group $\Bbb T$. Let the $A_n$ be a base of neighborhoods of $1$ in the subspace topology such that $A_{n+1}A_{n+1}^{-1}\subseteq A_n$ with $A_1$ small enough. Then no $A_n$ can contain a subgroup of the embedded $\Bbb Z$. Because every subgroup has to be dense.