Real number (as Cauchy sequence)

Written by Patrick Stevens, et al. last updated
Requires: Math 3

Consider the set of all Cauchy sequences of rational numbers: concretely, the set

Define an equivalence relation on this set, by if and only if, for every rational , there is a natural number such that for all bigger than , we have . This is an equivalence relation (exercise).

Show solution

  • It is symmetric, because .
  • It is reflexive, because for every , and this is .
  • It is transitive, because if for sufficiently large , and for sufficiently large , then for sufficiently large ; so by the triangle_inequality, for sufficiently large .

Write for the equivalence class of . (This is a slight abuse of notation, omitting the brackets that indicate that is actually a sequence rather than a rational number.)

The set of real numbers is the set of equivalence classes of under this equivalence relation, endowed with the following totally ordered field structure:

  • if and only if or there is some such that for all , .

This field structure is well-defined (proof).

Examples

  • Any rational number may be viewed as a real number, being the class (formally, the equivalence class of the sequence ).
  • The real number is indeed a real number under this definition; it is represented by, for instance, . It is also represented as , along with many other possibilities.