Question

Step 1

As per norms, the first question is answered: To prove that a closed subspace Y of a complete metric space X is complete.

Step 2

A metric space X is complete if all Cauchy sequences in X converge in X. A subspace Y of X is closed if it containes all the limit points of sequences in Y. We need to show that Y is complete in itself (in the induced topology form X); in other words, every Cauchy sequence in Y converges in Y.

Step 3

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