Solution
= Solution
If $\pi$ omits a prime $p$, then $A_\pi\subseteq A_{\neg p}\subseteq\mathbb Z_p$ by part i. The reductions
$$
\mathbb Z_p\longrightarrow\mathbb Z/p^n\mathbb Z
$$
separate its nonzero elements, so their restrictions separate the elements of $A_\pi$. Thus $A_\pi$ is <residually finite>. If $\pi$ contains every prime, then $A_\pi=\mathbb Q$, which has no nontrivial finite quotient because it is a <divisible group>.