Number System - Irrationals and Dedekind Cut

Dedekind cuts were the first, but they are not the most important construction of the reals. That would be Cauchy sequences.

A Cauchy sequence is a sequence that is converging. More precisely it is a sequence x_0, x_1, x_2, ... with the property that for any epsilon > 0, there exists N such that if n, m are both larger than N, then |x_n - x_m| < epsilon.

Two Cauchy sequences x and y are equivalent if they "should converge to the same spot". Technically this can be stated in the cute manner that x_0, y_0, x_1, y_1, x_2, y_x, ... is a Cauchy sequence.

With all of those definitions out of the way, a real number is defined to be an equivalence class of Cauchy sequences of rational numbers. It is straightforward to prove that you can define the usual algebraic operations on them, and they work out to the real numbers.

These two definitions look different, but are exactly equivalent. Why, then, would I claim that Cauchy sequences are more important than Dedekind cuts?

The answer is how you generalize them. Dedekind cuts depend on the existence of an ordering operation on the reals. Cauchy sequences only on the notion of distance. But there are a lot of mathematical objects of interest which have a notion of distance but usually not one of ordering. They are called metric spaces. The Cauchy sequence construction generalizes to a construction of the completion of any kind of metric space. The Dedekind cut construction pretty much only works for constructing reals out of rationals.