Sequentially Compact Metric Space is Compact/Proof 3
Jump to navigation
Jump to search
Theorem
A sequentially compact metric space is compact.
Proof
Follows directly from:
$\blacksquare$
Axiom of Countable Choice
This theorem depends on the Axiom of Countable Choice, by way of Countably Compact Metric Space is Compact.
Although not as strong as the Axiom of Choice, the Axiom of Countable Choice is similarly independent of the Zermelo-Fraenkel axioms.
As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.