An
ordered field F is said to be a
complete ordered field iff every nonempty
subset S of F which is
bounded above has a
supremum, or
least upper bound, in F.
The
real numbers are an example of a complete ordered field, while the
rational numbers are not (To see that the rationals are not a complete ordered field, consider the subset S := { rational numbers q such that q*q < 2 }. It is not difficult to see that the
least upper bound of this subset is sqrt(2), which
is not a rational number.)