Suppose $[\mathbb{Z}\_p:G] = n <\infty$. Write $n = p^km$ with $p\nmid m$. The idea is to show that $p^k\mathbb{Z}\_p = n\mathbb{Z}\_p \subseteq G$, after which I am done, since for any $x\in G$ we have that the open ball $x + p^k\mathbb{Z}\_p\subseteq G$, which makes $G$ open.

Unfortunately I am stuck at how to prove that $n\mathbb{Z}\_p\subseteq G$. Could someone lend a hand with a hint?