Let be a dense integral subdomain of the rational numbers and let be the positive terms of . As a result, is a directed type and a codirected type where the directed type operation is associative.
The Cauchy real numbers are inductively generated by the following:
The Cauchy real numbers are sometimes defined using sequences , with modulus of Cauchy convergence , respectively. However, the composition of a sequence and a modulus of Cauchy convergence yields a Cauchy approximation, so one could define Cauchy approximations as and as , with and following from function evaluation. In fact, Cauchy approximations and sequences with a modulus of Cauchy convergence are inter-definable.