The Real Analysis Lifesaver : : All the Tools You Need to Understand Proofs / / Raffi Grinberg.

• Superior document: Title is part of eBook package: De Gruyter Princeton University Press Complete eBook-Package 2017
• VerfasserIn: Grinberg, Raffi,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2017]; ©2017
• Year of Publication: 2017
• Language: English
• Series: Princeton Lifesaver Study Guides
• Physical Description: 1 online resource (200 p.) :; 33 line illus.

Real analysis is difficult. For most students, in addition to learning new material about real numbers, topology, and sequences, they are also learning to read and write rigorous proofs for the first time. The Real Analysis Lifesaver is an innovative guide that helps students through their first real analysis course while giving them the solid foundation they need for further study in proof-based math.Rather than presenting polished proofs with no explanation of how they were devised, The Real Analysis Lifesaver takes a two-step approach, first showing students how to work backwards to solve the crux of the problem, then showing them how to write it up formally. It takes the time to provide plenty of examples as well as guided "fill in the blanks" exercises to solidify understanding.Newcomers to real analysis can feel like they are drowning in new symbols, concepts, and an entirely new way of thinking about math. Inspired by the popular Calculus Lifesaver, this book is refreshingly straightforward and full of clear explanations, pictures, and humor. It is the lifesaver that every drowning student needs.The essential "lifesaver" companion for any course in real analysisClear, humorous, and easy-to-read styleTeaches students not just what the proofs are, but how to do them-in more than 40 worked-out examplesEvery new definition is accompanied by examples and important clarificationsFeatures more than 20 "fill in the blanks" exercises to help internalize proof techniquesTried and tested in the classroom

Table of Contents

Frontmatter -- Contents -- Preliminaries -- 1. Introduction -- 2. Basic Math and Logic* -- 3. Set Theory* -- Real Numbers -- 4. Least Upper Bounds* -- 5. The Real Field* -- 6. Complex Numbers and Euclidean Spaces -- Topology -- 7. Bijections -- 8. Countability -- 9. Topological Definitions* -- 10. Closed and Open Sets* -- 11. Compact Sets* -- 12. The Heine-Borel Theorem* -- 13. Perfect and Connected Sets -- Sequences -- 14. Convergence* -- 15. Limits and Subsequences* -- 16. Cauchy and Monotonic Sequences* -- 17. Subsequential Limits -- 18. Special Sequences -- 19. Series* -- 20. Conclusion -- Acknowledgments -- Bibliography -- Index