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.
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Henno Brandsma
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Clarkson
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1https://math.stackexchange.com/q/921022/4280 – Henno Brandsma Sep 30 '18 at 11:42
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There is an example on page 3 of these notes – Henno Brandsma Sep 30 '18 at 11:53
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see example $2.3.17$ in this book! – Chinnapparaj R Sep 30 '18 at 12:02
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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.
GEdgar
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