How to prove that every Cauchy sequence of elements in $\mathbb{Q}_p$ converges to an element in $\mathbb{Q}_p$?
Approach: I proved that this holds for $\mathbb{Z}_p$. But I don't know how to go from this to $\mathbb{Q}_p$. Can someone help me? Thanks!
Cauchy sequence: Take an arbitrary $\epsilon >0$. For every row $b_1,b_2, ...$ in $\mathbb{Q}_p$, there is an $N \in \mathbb{N}$ such that for every $i,j \geq N$: $d(b_i,b_j) < \epsilon$
Every Cauchy sequence is bounded in $\Bbb Q_p$. Each bounded set lies in $p^{-m}\Bbb Z_p$ for some $m$. So the Cauchy sequence is $(p^{-m}a_m)$ where $a_m\in\Bbb Z_p$. Then $(a_m)$ is also a Cauchy sequence, so convergent.