Corollary 2.53 Let (M, d) be a metric space and K C M. The following statements are equivalent: (i) K is relatively compact. (ii) K is totally bounded and (K,d) is complete.

Algebra & Trigonometry with Analytic Geometry
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ISBN:9781133382119
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Chapter10: Sequences, Series, And Probability
Section10.4: Mathematical Induction
Problem 46E
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prove corollary

Corollary 2.53 Let (M, d) be a metric space and K C M. The following statements
are equivalent:
(i) K is relatively compact.
(ii) K is totally bounded and (K,d) is complete.
Corollary 2.54 Let (M, d) be a complete metric space and K C M. The following
statements are equivalent:
(i) K is relatively compact.
(ii) K is totally bounded.
Proof:
excercises.
Transcribed Image Text:Corollary 2.53 Let (M, d) be a metric space and K C M. The following statements are equivalent: (i) K is relatively compact. (ii) K is totally bounded and (K,d) is complete. Corollary 2.54 Let (M, d) be a complete metric space and K C M. The following statements are equivalent: (i) K is relatively compact. (ii) K is totally bounded. Proof: excercises.
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