Cauchy Sequence - in Real Numbers

In Real Numbers

A sequence

of real numbers is called Cauchy, if for every positive real number ε, there is a positive integer N such that for all natural numbers m, n > N

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 to be infinitesimal for every pair of infinite m, n.