Instead of defining real numbers out of the blue simply postulating their existence and accepting their properties as axioms, like it is often done in literature, we prefer to construct them from rational numbers, which we have introduced in the previous part. In this chapter, we will prepare some technical prerequisites in order to do that. In particular, we want to show that real numbers can be defined as limits of certain sequences of rational numbers, and that these limits not only exist but also are unique, even though the rational sequence tending to this sequence is replaced by another one.
With this respect, we will find that real numbers can be defined as equivalence classes of sequences of rational numbers tending to the same limit. The technical prerequisites required for this interpretation of real numbers are rational Cauchy sequences, which we will introduce shortly. First, we start with explaining what rational sequences are, then what rational Cauchy sequences are and how we can add and multiply them with each other, in particular, that they form a ring. Finally, we will show that rational Cauchy sequences tending to $0$ are an ideal of this ring. This ideal will finally enable us to define real numbers.
