Let (M,d) be a metric space and let AC M be non-empty. Prove that the following...

 

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Let (M,d) be a metric space and let AC M be non-empty. Prove that the following are equivalent: (1) A is bounded. (2) There exists a D20 so that, for every a, b A, we have d(a, b) D. Problem 5: Let (M,d) be a metric space, let the sequence (yn) C M converge to a M. Show that (yn) is a Cauchy sequence. Problem 6: Let (M, d) be a metric space. Show that every Cauchy sequence in M is bounded. Let (M,d) be a metric space and let AC M be non-empty. Prove that the following are equivalent: (1) A is bounded. (2) There exists a D20 so that, for every a, b A, we have d(a, b) D. Problem 5: Let (M,d) be a metric space, let the sequence (yn) C M converge to a M. Show that (yn) is a Cauchy sequence. Problem 6: Let (M, d) be a metric space. Show that every Cauchy sequence in M is bounded.