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The real numbers can be characteriesed in two ways:

a) It is the unique ordered field with the least upper-bound property

b) It is the unique ordered archimedean field in which all cauchy sequences converge

Now both the least upper-bound property or the convergence of cauchy sequences can be taken as the idea of completeness. But why the difference? In a) we do not need to specify that it is archimedean, in b) we do.

Is this because the completeness in a) is due to order, whereas in b) its due to a metric $d(a,b):=|a-b|$?

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  • $\begingroup$ Archimedean is needed to exclude the p-adics. $\endgroup$ – Mark Bennet Jul 31 '13 at 16:36
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One way to look at the difference is to consider that the least upper-bound property is such a strong statement that it implies that an ordered field is Archimedean. If an ordered field has the least upper-bound property, then its subset of integers cannot have a field element $B$ as a supremum, because $B/2$ would be a lesser upper bound.

Convergence of Cauchy sequences is comparatively weak. When I saw your question, my (shameful) first instinct was not to really think about the problem, but to bust out James Propp's article Real Analysis in Reverse and look for a counterexample. And there it is: the field of formal Laurent series has convergence of Cauchy sequences, but not the least upper-bound property. The problem is that convergence of Cauchy sequences allows for fields that are too big.

I'm not sure if that answers your question about the fundamental difference between the two properties, but I hope it helps!

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