Show that every proper subgroup is finite

Question

For any prime $p$, show that every proper subgroup of the Prüfer $p$-group $(\{a/p^n+\mathbb{Z}:a, n \in \mathbb{Z}, n\geq0\}) $ is finite.

Given a subgroup $H$, I suppose this should have order $p^k$ for a $k$ maximum. Otherwise, $H$ must contain the Prüfer $p$-group, contradicting the fact that $H$ is proper. How can I write this proof using an appropriate notation?

Answer 1

First note $\frac{a}{p^k}$ with $(a,p)$ coprime generates all elements of the form $\frac{b}{p^k}$.

Assume $H$ is a proper subgroup, it must thus not contain an element, call it $\frac{b}{p^k}$ and now notice $H$ can only contain elements of the form $\frac{a}{p^l}$ with $l<k$, and is thus finite.