Suppose $\Bbb{Z}_p$ is complete as metric space, then, how can I prove $ \Bbb{Q}_p$(defined as fraction field of $\Bbb{Z}_p$) is also complete ?
My try
Suppose {$a_n$} be caushy sequence in $\Bbb{Q}_p$. {$p^na_n$} is Cauchy sequence in $\Bbb{Z}_p$ for some $n$. {$p^na_n$} has limit point $α$ in $\Bbb{Z}_p$, hence {$a_n$} converges to $α/p^n$. So,$\Bbb{Q}_p$ is complete.