There is an exercise telling that every finite subset of group ($\mathbb Q$,+) or of group $\mathbf{Z}_{p^\infty}$ generates a cyclic group itself.
For the first group if $X= \left\{\frac{p_{1}}{q_{1}},\frac{p_{2}}{q_{2}},\ldots,\frac{p_{n}}{q_{n}}\right\} $ be a finite subset, then obviously $X\subseteq \langle\frac{1}{q_{1} q_{2}...q_{n}}\rangle$ and so $\langle X\rangle$ is cyclic iself.
Kindly asking about the second group. How to show that about $\mathbf{Z}_{p^\infty}$ ? Thanks.