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How to prove that a closed subspace of a Banach space is a Banach space?

A subspace is closed if it contains all of its limit points. But in the proof of the above question how can use this idea to get a Cauchy sequence and show that it is convergent in the subspace?

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    You don't need to get a Cauchy sequence. You just assume a sequence is Cauchy and prove it converges and the limit point is in the subspace.2012-11-19

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