Compact Metric Space is Complete
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Theorem
A compact metric space is complete.
Proof
Follows directly from:
- Compact Space is Countably Compact
- Countably Compact Metric Space is Sequentially Compact
- Sequentially Compact Metric Space is Complete
$\blacksquare$
Sources
- 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $3.11 b$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces