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exercise 5 based on tao let xd be an arbitrary...

Question: exercise 5 based on tao let xd be an arbitrary...

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Exercise 5 (based on Tao). Let (X,d) be an arbitrary metric space. Prove the following statements (1) If a sequence is convergent in X, all its subsequences are converging to the same limit as the original sequence. (2) If a subsequence of a Cauchy sequence is convergent, then the whole sequence is convergent to the same limit as the subsequence. (3) Suppose that (X,d) is complete and Y S X is closed in (X,d). Then the space (Y,dlyxy) is complete. (4) Give two examples of metric spaces (with explanations), where every subset of the original space is bounded.

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