The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs (Princeton Lifesaver Study Guides)

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 The essential lifesaver that every student of real analysis needs 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 "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."-- Fourni par l'éditeur 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 2lifesaver3 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 2fill in the blanks3 exercises to help internalize proof techniques Tried and tested in the classroom "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." -- Publisher's description Cover 1 Title 4 Copyright 5 CONTENTS 6 Preliminaries 8 1 Introduction 10 2 Basic Math and Logic* 13 3 Set Theory* 21 Real Numbers 32 4 Least Upper Bounds* 34 5 The Real Field* 42 6 Complex Numbers and Euclidean Spaces 53 Topology 66 7 Bijections 68 8 Countability 75 9 Topological Definitions* 86 10 Closed and Open Sets* 97 11 Compact Sets* 105 12 The Heine-Borel Theorem* 115 13 Perfect and Connected Sets 124 Sequences 134 14 Convergence* 136 15 Limits and Subsequences* 145 16 Cauchy and Monotonic Sequences* 155 17 Subsequential Limits 164 18 Special Sequences 173 19 Series* 181 20 Conclusion 190 Acknowledgments 194 Bibliography 196 Index 198 Content: Cover Title Copyright 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.