Proof Review: Every compact set is complete

Question

$X$ is a compact metric space. Let $A \subseteq X$ be a compact set. Let $\{C_n\} \in A$ be a Cauchy sequence. Because it is Cauchy, $\{C_n\}$ must converge somewhere (although it doesn't need to converge in $A$). Let the limit point for $\{C_n\}$ be denoted by $C$. We also have that $A$ is compact $\implies A$ is closed and bounded $\implies$ $C \in A$ (because closed sets include the limit points for all its convergent sequences) $\implies$ A is complete.

Answer 1

Sorry, but if you say that the Cauchy sequence must converge somewhere, you're already using completeness, which you are to prove.

What you know, instead, is that the sequence has a cluster point, because of compactness. But a Cauchy sequence with a cluster point is convergent.

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Note that, in general metric spaces, closed and bounded sets need not be compact (the converse is true, but of little use, generally). However you used the converse implication (compact implies closed).