This graduate thesis is divided into five major thematic chapters through which the construction of a set of real numbers, that is, a completely ordered field, is shown. The first chapter contains the definitions of basic notions. These are the notions of binary operations and binary relations on a nonempty set and their properties, as well as the concepts of various algebraic structures. In the second chapter we give the axioms of natural numbers, we define addition and multiplication on the set of natural numbers, we prove their properties and construct an order on the same set. The set of integers, addition, multiplication and the order on it are defined in the third chapter. The fourth chapter contains the construction of the set of rational numbers and the aforementioned concepts with the additional concept of the absolute value of a rational number. The fifth chapter begins with the concept of a sequence and its convergence. With the definition of Cauchy sequence in the set of rational numbers and the relation of equivalence at the Cauchy sequence, a definition of a set of real numbers comes up. Thereafter, along with the construction of the addition, multiplication and the order on the set of real numbers, through Cauchy sequences in that set and the proof that they are convergent, comes the ultimate concept of a completely ordered field.