Given a sequence a_1, a_2, ..., a_n, ..., we say it is a Cauchy sequence if for every positive e (epsilon), there exists a (positive) integer N such that |a_i - a_j| < e for all i > N and all j > N.

As the set of rational numbers is not complete, there exist Cauchy sequences of rational numbers that do not converge to rational numbers. In other words, their limits are irrational numbers.

Thus, if you want to construct the real numbers (in particular the irrational numbers), you may define them as Cauchy sequences of rational numbers. (Intuitively speaking, we define the real numbers as "limits" of Cauchy sequences.