MATHEMATICAL ANALYSIS I

This book is organized as follows. We first recall the real number system with some basic properties in Chapter 1. The Axiom of Completeness is a key property of this subject. Chapter 2 deals with a sequence of real numbers. The definition of a convergentsequence is given together with several important properties. Two interesting theorems,namely the Monotone Convergence Theorem and the Bolzano–Weierstrass Theorem arediscussed. A criterion for the test of convergence of a sequence of real numbers is given in terms of being a Cauchy sequence. At the end of this chapter, we introduce the concept of an infinite limit of a sequence. The concept of continuous functions is given in Chapter 3. We show that the continuity of a function is closely related to the convergence of a sequence and hence many properties of continuous functions are deduced. We also discuss the definition of a limit of functions and an equivalent formulation in terms of a convergent sequence. Chapter 4 focuses on the differentiable functions. The Mean Value Theorem is given and applied to prove the well known L’Hˆopital’s Rule. The integrable functions are studied in Chapter 5. We recall and prove two fundamental theorems of calculus. Some properties about integrable functions are discussed.