Yau Donald. A First Course in Analysis

World Scientific, 2013. — 206 p.This book is an introductory text on real analysis for undergraduate students. The prerequisite for this book is a solid background in freshman calculus in one variable. The intended audience of this book includes undergraduate mathematics majors and students from other disciplines who use real analysis. Since this book is aimed at students who do not have much prior experience with proofs, the pace is slower in earlier chapters than in later chapters. There are hundreds of exercises, and hints for some of them are included.This is an introductory text on real analysis for undergraduate students. The first course in real analysis is full of challenges, both for the instructors and the students.
Many mathematics majors consider real analysis a difficult course. The transition from mechanical computation to formal, rigorous proofs is difficult even for many mathematics majors. Most students beginning a course in real analysis have never been asked to understand and construct proofs before. Moreover, even if one has some ideas about how a proof should go, writing it down in a logical manner is a challenge in itself. This book is written with these challenges in mind.
The prerequisite for this book is a solid background in freshman calculus in one variable. The intended audience of this book includes undergraduate mathematics majors and students from other disciplines whose use real analysis. Since this book is aimed at students who do not have much prior experience with proofs, the pace is slower in earlier chapters than in later chapters.
In most instances, motivations for new concepts are explained before the actual definitions. For many concepts that have negations (for example, convergence of a sequence), such negations are also stated explicitly. Wherever appropriate we discuss the basic ideas that lead to a proof before the actual proof is given. Such discussion is intended to help students develop an intuition as to how proofs are constructed. There are exercises at the end of each section and of each chapter.
Occasionally, some further topics are explored in these additional exercises.