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Metric Spaces, Convergence, and Completeness
Metric Spaces, Convergence, and Completeness
3 questions
Difficulty 4-5
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A metric space is complete if every Cauchy sequence converges. Why does this matter?
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A.
Complete metric spaces are always compact, which is useful for extreme-value theorems
B.
Completeness lets you construct limits using the sequence alone, without knowing the limit in advance
C.
Completeness is equivalent to the axiom of choice, so it follows automatically in ZFC set theory
D.
Completeness means every function on the space is continuous by topological structure
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