The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs

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 them―in 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

Table of Contents

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