Construcão dos números reais por sequências de Cauchy e cortes de Dedekind

Abstract

We will study the construction of real numbers by two different methods. Firstly by Cauchy sequences, which is a shrewd and beautiful way of characterizing real numbers given our intuitive notion that such numbers can be used to represent points on a line, yet it is also possible to prove all the usual properties of these numbers. This construction is essentially done via an equivalence relationship established in the Cauchy’s set of rational sequences with the initial hypothesis that the ordered body of rational numbers is already known. The construction made by Dedekind cuts is essentially different, because in the place of the language of sequences it is used the language of sets, although the diference of language arrives at the same results on the algebraic properties of these sets. Finally we will see that there is only one complete ordered body, up to isomorphisms.