Editing Cauchy sequence (section)

==In real numbers==
A sequence
<math display="block">x_1, x_2, x_3, \ldots</math>
of real numbers is called a Cauchy sequence if for every [[Positive and negative numbers|positive]] real number <math>\varepsilon,</math> there is a positive [[integer]] ''N'' such that for all [[natural numbers]] <math>m, n > N,</math>
<math display="block">|x_m - x_n| < \varepsilon,</math>
where the vertical bars denote the [[absolute value]].  In a similar way one can define Cauchy sequences of rational or [[complex number]]s.  Cauchy formulated such a condition by requiring <math>x_m - x_n</math> to be [[infinitesimal]] for every pair of infinite ''m'', ''n''.

For any real number ''r'', the sequence of truncated decimal expansions of ''r'' forms a Cauchy sequence.  For example, when <math>r = \pi,</math> this sequence is (3, 3.1, 3.14, 3.141, ...).  The ''m''th and ''n''th terms differ by at most <math>10^{1-m}</math> when ''m'' < ''n'', and as ''m'' grows this becomes smaller than any fixed positive number <math>\varepsilon.</math>

===Modulus of Cauchy convergence===

If <math>(x_1, x_2, x_3, ...)</math> is a sequence in the set <math>X,</math> then a ''modulus of Cauchy convergence'' for the sequence is a [[Function (mathematics)|function]] <math>\alpha</math> from the set of [[natural number]]s to itself, such that for all natural numbers <math>k</math> and natural numbers <math>m, n > \alpha(k),</math> <math>|x_m - x_n| < 1/k.</math>

Any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The existence of a modulus for a Cauchy sequence follows from the [[well-ordering property]] of the natural numbers (let <math>\alpha(k)</math> be the smallest possible <math>N</math> in the definition of Cauchy sequence, taking <math>\varepsilon</math> to be <math>1/k</math>). The existence of a modulus also follows from the principle of [[countable choice]]. ''Regular Cauchy sequences'' are sequences with a given modulus of Cauchy convergence (usually <math>\alpha(k) = k</math> or <math>\alpha(k) = 2^k</math>). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent to a regular Cauchy sequence; this can be proven without using any form of the axiom of choice.

Moduli of Cauchy convergence are used by constructive mathematicians who do not wish to use any form of choice. Using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis. Regular Cauchy sequences were used by {{harvtxt|Bishop|2012}} and by {{harvtxt|Bridges|1997}} in constructive mathematics textbooks.