Every finite dimensional normed space (over a complete field, namely or ) is Banach (complete in the metric induced by the norm).

**Proof**: Let be an n-dimensional normed space with basis and let be any Cauchy sequence in . As is Cauchy, for any given there exists an such that for all we have . However, by a previous lemma,there exists a such that

Which means that , which in turn implies for a fixed , the sequence is Cauchy in a complete space, so it converges. Let the limit of this sequence be and let . Now we have

.

Since , we can make this right-hand side arbitrarily small by picking a large enough and considering . This means that , so , and the space is complete.

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