Problem: Let $A_n=\{\frac{m}{n}:m\in\mathbb{Z}\}$. Find $\limsup_{n\rightarrow\infty}A_n$ and $\liminf_{n\rightarrow\infty}A_n$
Attempted: It is clear that limsup should be $\mathbb{Q}$. I can show it using the "infinitely often" definition of limit superior and double inclusion technique. However, I am a bit confused about how do I find the liminf, which I believe should be $\mathbb{Z}$. To be specific, I don't know how to show $\liminf_{n\rightarrow\infty}A_n\subseteq\mathbb{Z}$.
Suppose that $q\in\Bbb Q\setminus\Bbb Z$. Write $q$ in lowest terms as $q=\frac{m}n$ with $n\ge 2$. Show that for each $k\in\Bbb Z^+$, $q\notin A_{kn+1}$. (In fact, $q\in A_k$ if and only if $n\mid k$.) Thus, $q$ is not eventually in $A_k$, and hence $q\notin\liminf_kA_k$.
The argument that you sketch for $\limsup_kA_k=\Bbb Q$ is correct.