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 $d(a,b):=|a-b|$?