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.)