Prove that the subspace.of a complete metric is complete if and only if it is closed
Let us prove that the subspace of a complete metric is complete if and only if it is closed.
Let be complete and Then there exists a sequence converging to . Obviously, this sequence is a Cauchy sequence, and, since is complete, it converges to some . Since the limit of a sequence is unique in a metric space, we see that
Let be closed and be a Cauchy sequence in . Since is complete, converges to some . But as is closed, has to be in .
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