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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 analysis
- Clear, humorous, and easy-to-read style
- Teaches students not just what the proofs are, but how to do themin more than 40 worked-out examples
- Every new definition is accompanied by examples and important clarifications
- Features more than 20 “fill in the blanks” exercises to help internalize proof techniques
- Tried and tested in the classroom
About the Author
Raffi Grinberg is an entrepreneur and former management consultant. He graduated with honors from Princeton University with a degree in mathematics in 2012.
Table of Contents
1 Introduction 3
2 Basic Math and Logic* 6
3 Set Theory* 14
Real Numbers 25
4 Least Upper Bounds* 27
5 The Real Field* 35
6 Complex Numbers and Euclidean Spaces 46
7 Bijections 61
8 Countability 68
9 Topological Definitions* 79
10 Closed and Open Sets* 90
11 Compact Sets* 98
12 The Heine-Borel Theorem* 108
13 Perfect and Connected Sets 117
14 Convergence* 129
15 Limits and Subsequences* 138
16 Cauchy and Monotonic Sequences* 148
17 Subsequential Limits 157
18 Special Sequences 166
19 Series* 174
20 Conclusion 183