Is it true that a metric compact is complete space?

Question

Is it true that a metric compact is complete space? I think that it is true.

Answer 1

A compact metric space $X$ is sequentially compact. Thus any Cauchy sequence in $X$ has a convergent subsequence. But in a metric space, this is equivalent to the entire sequence being convergent.

Answer 2

For more details see: Joshi, Introduction To General Topology, p282.