In this chapter, we will first construct a model of the real numbers using Cauchy sequences of rational numbers. We also present a second model of the real numbers according to A’Campo [1]. This construction has the advantage that it only relies on the integers and not on the rational numbers, and that the definition of the multiplication is much simpler and natural than the classical definition based on equivalence classes of Cauchy sequences. Afterwards, we will show that both constructions yield isomorphic models. The constructions of the real numbers will be quite general, such that— depending on whether we start with the standard or a non-standard model of the natural numbers—we obtain the standard or a non-standard model of the real numbers.