Introduction to Real Analysis

This is a text that develops calculus 'from scratch', with complete rigorous arguments. Its aim is to introduce the reader not only to the basic facts about calculus but, as importantly, to mathematical reasoning. It covers in great detail calculus of one variable and multivariable calculus. Additionally it offers a basic introduction to the topology of Euclidean space. It is intended to more advanced or highly motivated undergraduates.
Contents: The Basics of Mathematical ReasoningThe Real Number SystemSpecial Classes of Real NumbersLimits of SequencesLimits of FunctionsContinuityDifferential CalculusApplications of Differential CalculusIntegral CalculusComplex Numbers and Some of Their ApplicationsThe Geometry and the Topology of Euclidean SpacesContinuityMulti-variable Differential CalculusApplications of Multi-variable Differential CalculusMultidimensional Riemann IntegrationIntegration over Submanifolds
Readership: More advanced undergraduate students and professionals who is interested in calculus and mathematical analysis. Real Analysis;Differential Calculus;Integral Calculus;Integration Over Submanifolds0Key Features:Emphasis on rigorous arguments early on. It assumes the reader has little or no experience with mathematical proofs and it introduces the reader to the basics of this process: basic aristotelian logic and foundations of the number systemMany detailed classical nontrivial examples. It is the belief of the reader that the modern students is not exposed enough to the classical examples that marked and promoted the development of analysis. We tried to address this by including many such examples, some quite elaborate developed gradually over several chaptersUnlike the traditional American texts, the key concept of limit is introduced via through sequences. This presents the concept in the cleanest form, free of extraneous details and has a further payoff when dealing with the topology of Euclidean spacesAll the results in the book have complete proofs. However, it is not realistic to assume that all these proofs can be included during class without hurting the understanding. To help the instructor, I wrote in fine print the proofs that can be skipped without affecting the overall understanding. The decisions which proofs to present with fewer details are based on the experience of teaching each of the two logical parts of the book for three consecutive years