Closed subgroups of $\mathbb Q_p$

Question

Let $\mathbb Q_p$ be the field of $p$-adic numbers, $\mathbb Z_p$ the ring of $p$-adic integers.

Is there a closed subgroup of $\mathbb Q_p$ other than the following list?

1) 0

2) $p^n\mathbb Z_p$, $n\in \mathbb Z$.

3) $\mathbb Q_p$

I think I came up with sort of a proof that the answer is no, but I'm not 100% sure.

Answer 1

If a subgroup contains $x\ne 0$ then it contains $x\Bbb Z$, and if it's closed then it contains the limit points all collected in $x\Bbb Z_p$, but we also know $x\Bbb Z_p=p^{v_p(x)}\Bbb Z_p$. We can union over all $x$ to get the subgroup.

Therefore, every nonzero closed subgroup is a union of $p^r\Bbb Z_p$s, but these are linearly ordered under inclusion, with $\Bbb Q_p$ the union of any infinite collection of them. So the list is exhaustive.