Equivalence of Definitions of Complete Metric Space
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Theorem
The two definitions of complete metric space:
- $(1):\quad$ A metric space $\left({X, d}\right)$ is complete if every Cauchy sequence is convergent
- $(2):\quad$ A metric space $\left({X, d}\right)$ is complete iff the intersection of every nested sequence of closed balls whose radii tend to zero is non-empty
are logically equivalent.
Proof
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Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$: Complete Metric Spaces
