Compact Metric Space is Complete

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
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