Show that $\mathbb{Q_p} $ is locally compact Suppose $\mathbb{Q_p} $ is the fraction field of $\mathbb{Z_p}$ ($p$-adic integers) i.e.

$$\mathbb{Q_p} = \left\lbrace\frac{x}{y} \space \bigg{|} \space x,y \in \mathbb{Z_p} , y\neq 0 \right\rbrace$$ 

Now with respect to the topology defined by $d(x,y) = e^{-v_p(x-y)}$ ($v_p$ is the $p$-adic valuation) , we need to show that $\mathbb{Q_p} $ is locally compact.
Any suggestions?

The accepted answer is not complete because it does not argue why $p^n\mathbb{Z}_p$ is compact. I think the following idea deal with this:
To prove that $\mathbb{Q}_p$ is locally compact is enough to prove that $\mathbb{Z}_p$ is compact. For this, thinking in the $p$-adic integers as sequences we have $$\mathbb{Z}_p=\{(a_0,a_1,a_2,...)\mid a_n\in \mathbb{Z}/p\mathbb{Z} \}=(\mathbb{Z}/p\mathbb{Z})^\mathbb{N}$$
Now, by Tychonoff  $(\mathbb{Z}/p\mathbb{Z})^\mathbb{N}$ is compact with the product topology and also is easy to see that $x+p^n\mathbb{Z}_p$ is an open set of this topology. So every cover of $\mathbb{Z}_p$ with sets of the form $x+p^n\mathbb{Z}_p$ will have a finite subcover and then $\mathbb{Z}_p$ is compact.
