How to prove that $(\mathbb{Q},d_p)$ is not complete?

Question

How do we prove that the metric space $(\mathbb{Q},d_p)$, with $d_p$ the $p$-adic distance, is not complete ? Can anyone construct a Cauchy sequence that does not converge? Thank you very much in advance.

Answer 1

Example. The series $$ \sum_{k=0}^\infty p^{k!} $$ represents an irrational $p$-adic number.

More generally, consider the $p$-adic expansion $$ x = \sum_{j=n}^\infty s_j p^j $$ where each $s_j \in \{0,1,.\dots,p-1\}$. Then $x$ is rational if and only if the sequence $s_j$ is eventually periodic. For the proof, follow the usual proof for rationality of base $10$ decimals, with appropriate easy changes.