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

Preliminaries 1 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 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