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Why is locally compact abelian normed group always complete? It seems to be something very simple, but I'm unable neither to prove it, nor find any references in the literature.

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By local compactness, there is an $r > 0$ so small that the ball of radius $r$ centered at the origin is precompact. Hence every ball of radius $r$ is precompact. Now given a Cauchy sequence, all but finitely many terms are contained in some ball of radius $r$...

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