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.



locally compact abelian normed group
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 $r$...


