Real numbers may be defined as

equivalence classes of

Cauchy sequences of

rational numbers. Two

Cauchy sequences *S*_{1} = {(

*n*,

*a*_{n})} ,

*S*_{2} = {(

*n*,

*b*_{n})} of

rational numbers are equivalent if and only if their intertwining sequence,

*S*_{3} = (

*S*_{1} ,

*S*_{2}) = {(

*2n-1*,

*a*_{n})} U {(

*2n*,

*b*_{n})} = {(

*n*,

*c*_{n})} is also a

Cauchy sequence of

rational numbers.

That is to say, if for every natural number *r* there exists a natural number *N* such that for all natural numbers *n* and for all natural numbers *m*, if *n*, *m* > *N*, then | *c*_{n} - *c*_{m} | < 1/*r*, then the two Cauchy sequences are equivalent.

Note that for {(*n*, *c*_{n})} , *c*_{n} = *a*_{(n+1)/2} , if *n* is odd and *c*_{n} = *b*_{n/2} , if *n* is even, i.e., *c*_{1} = *a*_{1} , *c*_{2} = *b*_{1} , *c*_{3} = *a*_{2} ,

*c*_{4} = *b*_{2} , ...

The equivalence class of a Cauchy sequence {(*n*, *a*_{n})} of rational numbers is denoted by [{(*n*, *a*_{n})}] , and for any other Cauchy sequence {(*n*, *b*_{n})} of rational numbers , if {(*n*, *b*_{n})} is equivalent to {(*n*, *a*_{n})} ,

then [{(*n*, *b*_{n})}] = [{(*n*, *a*_{n})}].

The definition then is, for any Cauchy sequence {(*n*, *a*_{n})} of rational numbers , **R** = [{(*n*, *a*_{n})}] is a real number.

For any rational number *a*, the sequence {(*n*, *a*)} is obviously a Cauchy sequence and its equivalence class is denoted by either [{(*n*, *a*)}] or **phi**(*a*), which is also the set of all Cauchy sequences that converge to *a*, i.e., have limit *a*. A real number is said to be irrational if it is not in the range of **phi**. That is, a real number **R** is irrational if there does not exist an equivalence class of a Cauchy sequence {(*n*, *a*)} of rational numbers such that **R** = [{(*n*, *a*)}] = **phi**(*a*) = { *S* | *S* converges to *a* }. Otherwise, a real number **R** is said to be rational.

This definition, due to Georg Cantor, allows one to construct the real numbers from the rational numbers in a manner that forms a complete ordered field. Note that here, *completeness* for real numbers is taken to be that every Cauchy sequence of *real numbers* (i.e., every Cauchy sequence of equivalence classes of Cauchy sequences of rational numbers) converges to a real number (equivalence class of a Cauchy sequence of rational numbers). This formulation of the notion of completeness is equivalent to the statement that every non-empty subset of real numbers that has an upper bound (lower bound) has a supremum (infimum) that is also a real number, which is what distinguishes the reals from the rationals.

This method of constructing the real numbers from the rational numbers as well as that of Dedekind cuts provide the only truly rigorous developments of the real number system. Cantor's method is preferred, however, because it can also be applied to measure theory.

Proofs provided upon request.