The essential "lifesaver" that every student of real analysis needsReal 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.

Preliminaries 11 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
Topology 59
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
Sequences 127
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
Acknowledgments 187
Bibliography 189
Index 191