If $G\cong \mathbb{Z}\times \mathbb{Z}\times \dots \times \mathbb{Z}$ is a finitely generated abelian group without torsion of rank $n$, where $n$ is the number of copies of $\mathbb{Z}$. Then any subgroup $H$ in $G$ is also finitely generated without torsion of rank $m$ where $m\leq n$
It looks clear, but how can we show this? What are the possible approaches to this question?