Is the following descending sequence nonzero?

Question

Let $K_{1}\supseteq K_{2}\supseteq K_{3}\supseteq \cdots$ be a descending sequence of compact subgroups of compact, torsion-free group $G$. Is $\bigcap_{r=1}^\infty n^r K\neq 0$? (for a positive integer $m$, $mK=\{mx;x\in K\}$)

Answer 1

It seems that the answer is negative. Let $\mathbb{Z}_p$ be the group of $p$-adic integers, which should be compact and torsion-free. Then $\mathbb{Z}_p\supset p\mathbb{Z}_p\supset p^2\mathbb{Z}_p\supset\dots$ and $\bigcap_{k=1}^\infty p^k\mathbb{Z}_p=\{0\}$, isn't it?