Sequences and Series of Real Numbers

The elementary theories of real-valued functions of a real variable and of numerical sequences and series are treated in any standard calculus text. In most cases, however, the proofs are given in appendices and omitted from the main body of the course. To give rigorous proofs of the basic theorems on convergence, continuity, and differentiability, one needs a precise definition of real numbers. One way to achieve this is to start with the construction of real numbers from the rational ones by means of Dedekind Cuts. We shall not follow this path. Instead, we will give a set of axioms for the real numbers from which all their properties can be deduced. These axioms will be divided into three categories: First, we introduce the algebraic ones. Next, we discuss the order axioms, and finally, we discuss the very deep and fundamental completeness axiom. After outlining the axiomatic definition of the real numbers, we will look at sequences in ℝ and their limits. Here, the most important concept is that of a Cauchy sequence. It will be used in Appendix A for a brief discussion of Cantor’s construction of real numbers from the Cauchy sequences in the set ℚ of rational numbers. The properties of sequences will be used in a short section on infinite series of real numbers. We shall return to infinite series in another chapter to discuss series of functions, such as power series and Fourier series. Finally, the last section is a brief introduction to unordered series and summability. Throughout this chapter, our universal set will be U = ℝ, so that a set will automatically mean a subset of ℝ.