Completeness and the Completeness Axiom

Question

Using the definition, a metric space is complete if every Cauchy sequence converges in the metric space.

The Completeness Axiom states that every nonempty set of reals that is bounded above has a supremum.

How are these two related? I'm having trouble seeing why one might imply the other, or if they are equivalent.

Answer 1

Wikipedia discusses several forms of completeness for the real numbers, ending with:

For an ordered field, Cauchy completeness is weaker than the other forms of completeness on this page. But Cauchy completeness and the Archimedean property taken together are equivalent to the others.