Positivity of Real Numbers

Since a real number is defined as a Cauchy sequence, it is tempting to say that a real number is positive if all . This definition, however, makes positive even if it converges to 0. Thus, we need a better definition.

Definition

A real number is positive iff there is a positive rational such that .

The definition of negative number is the same.

And a real number that is neither positive nor negative is 0.