ABSTRACT
In the previous chapter we used some very powerful results about sequences, such as the Monotone Convergence Theorem (Theorem 1.4.7) or Cauchy’s Test (Theorem 1.6.6). In this chapter, our goal is to prove these theorems. When Cauchy did that in Cours d’analyse he took some properties of real numbers as self-evident. In the course of the 19th century it became clear that these needed to be proved as well, and for that it was necessary to make a precise definition of real numbers. This task was accomplished around 1872 by the independent efforts of Dedekind, Cantor, Heine, and Me´ray.
