Preface Our goal in this set of lecture notes is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs. The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. In addition, the notes include many carefully selected exercises of various levels of difficulty. Hints and solutions to selected exercises are available in the back of the book. For each section, there is at least one exercise with hints or fully solved. For those exercises, besides the solutions, there are explanations about the process itself and examples of more general problems where the same technique may be used. Exercises with solutions are indicated by a ▶and those with hints are indicated by a ▷. The last chapter contains additional topics. These include topological properties of the real line, generalizations of the extreme value theorem and more contemporary topics that expand on the notions of continuity or optimization. Lower and upper semicontinuity, differentiation of convex functions, and generalized differentiation of non-differentiable convex functions can be used as optional mathematical projects. Finally, to make it easier for students to navigate the text, the electronic version of these notes contains many hyperlinks that students can click on to go to a definition, theorem, example, or exercise at a different place in the notes. These hyperlinks can be easily recognized because the text or number is on a different color and the mouse pointer changes shape when going over them. Changes in the Third Edition This third edition includes a number of improvements based on recommendations from students and colleagues and on our own experience teaching the course over the last several years. We reorganized the narrative in multiple sections. More significantly, the first four chapters now offer a streamlined presentation of the main topics without going into more abstract topological properties of the real line. While various definitions such as limits and continuity are defined in some
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