One of the important problems in analysis is determine how to represent real numbers. The natural definition which occurs from order theory is that a real number is defined by a Dedekind cut which is effectively a partition of the rational numbers into upper and lower sets relative to the real number. This defines the order theoretic properties of the real number. In order to make sense of the Dedekind cut, it is necessary to have a lower bound sequence and an upper bound sequence, the two of which together give an increasingly accurate interval approximation of the real number. The representation of a real number in terms of a sequence of intervals is equivalent to using two sequences anyways.
On the other hand, if we take the metric definition rather then the order definition we get Cauchy sequences which are sequences that satisfy the Cauchy criterion which states that consecutive numbers get increasing close together as the sequence goes on. We can represent any given number then using a cauchy sequence of rational numbers. In an implementation sense, this can be defined using a function between the set of natural numbers and the set of rational numbers, both of which are countable so this can be implemented computationally. A single sequence may not be enough to compute the order properties of the number, but it is enough to define the number in terms of the metric space.
