In a locally compact nontrivially valued field, the valuation ring is compact. A Cauchy sequence has a tail inside a translate of this compact set, hence a convergent subsequence. The Cauchy property forces the full sequence to converge.

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  1. Local compactness criterion for a complete non-Archimedean field
  2. Local field
  3. Non-Archimedean analysis
  4. Arithmetic
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