A dense additive subgroup of a complete metrizable topological group cannot be proper if it is comeagre. Every translate is comeagre and meets the subgroup, so every translating element is a difference of two subgroup elements. Applied to a topologically complete normed space inside its completion, this proves that its original norm is complete.

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  1. Baire category theorem
  2. Complete metric space
  3. Metric space
  4. Topological analysis
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