Question 1. [6 marks] Call a metric space sequentially compact if every sequence has a convergent subsequence. Prove tha

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Question 1 6 Marks Call A Metric Space Sequentially Compact If Every Sequence Has A Convergent Subsequence Prove Tha 1 (79.96 KiB) Viewed 39 times

Question 1. [6 marks] Call a metric space sequentially compact if every sequence has a convergent subsequence. Prove that a metric space is sequentially compact if and only if every infinite subset has a cluster point. Question 2. [6 marks] Call a metric space totally bounded if, for every e > 0, the metric space is the union of a finite number of closed balls radius ε. Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence. Question 3. [5 marks] Prove that a subset of a metric space is closed if and only if it contains all its cluster points. Question 4. [5 marks] Let S be a subset of the metric space M = (X,d), with point pe X. Prove that p is a cluster point of S if and only if p is the limit of a Cauchy sequence in Sn{p}.