Foundations of Mathematical Real Analysis

This book is intended as a serious introduction to the study of mathematical analysis. In contrast to calculus, mathematical analysis does not involve formula manipulation, memorizing integrals or applications to other fields of science. No. It involves geometric intuition and proofs of theorems. It is pure mathematics! Given the mathematical preparation and interest of our intended audience which, apart from mathematics majors, includes students of statistics, computer science, physics, students of mathematics education and students of engineering, we have not given the axiomatic development of the real number system. However, we assume that the reader is familiar with sets and functions. This book is divided into two parts. Part I covers elements of mathematical analysis which include: the real number system,bounded subsets of real numbers, sequences of real numbers,monotone sequences, Bolzano-Weierstrass theorem, Cauchy sequences and completeness of R, continuity, intermediate value theorem, continuous maps on [a, b], uniform continuity,closed sets, compact sets, differentiability, series of nonnegative real numbers, alternating series, absolute and conditional convergence; and re-arrangement of series. The contents of Part I are adequate for a semester course in mathematical analysis at the 200 level. Part II covers Riemann integrals. In particular, the Riemann integral, basic properties of Riemann integral, pointwise convergence of sequences of functions, uniform convergence of sequences of functions,series of real-valued functions: term by term differentiation and integration; power series: uniform convergence of power series; uniform convergence at end points; and equi-continuity are covered. Part II covers the standard syllabus for a semester mathematical analysis course at the 300 level. The topics covered in this book provide a reasonable preparation for any serious study of higher mathematics. But for one to really benefit from the book, one must spend a great deal of time on it, studying the contents very carefully and attempting all the exercises, especially the miscellaneous exercises at the end of the book. These exercises constitute an important integral part of the book. Each chapter begins with clear statements of the most important theorems of the chapter. The proofs of these theorems generally contain fundamental ideas of mathematical analysis. Students are therefore encouraged to study them very carefully and to discover these ideas.