information:highlighting, structural:expansion, structural:reformatting
In real numbers

A sequence x 1, x 2, x 3, … {\displaystyle x_{1},x_{2},x_{3},\ldots } of real numbers is called a Cauchy sequence if for every positive real number ε, {\displaystyle \varepsilon,} there is a positive integer N such that for all natural numbers m, n > N, {\displaystyle m,n>N,} | x m − x n | < ε, {\displaystyle |x_{m}-x_{n}|<\varepsilon,} where the vertical bars denote the absolute value. In a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring x m − x n {\displaystyle x_{m}-x_{n}} 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 r = π, {\displaystyle r=\pi,} this sequence is (3, 3.1, 3.14, 3.141,...). The m th and n th terms differ by at most 10 1 − m {\displaystyle 10^{1-m}} when m < n, and as m grows this becomes smaller than any fixed positive number ε. {\displaystyle \varepsilon.}