Cover -- Contents -- Symbols -- Introduction -- Part 1 Studying Analysis -- 1 What is Analysis Like? -- 2 Axioms, Definitions and Theorems -- 2.1 Components of mathematics -- 2.2 Axioms -- 2.3 Definitions -- 2.4 Relating a definition to an example -- 2.5 Relating a definition to more examples -- 2.6 Precision in using definitions -- 2.7 Theorems -- 2.8 Examining theorem premises -- 2.9 Diagrams and generality -- 2.10 Theorems and converses -- 3 Proofs -- 3.1 Proofs and mathematical theories -- 3.2 The structure of a mathematical theory -- 3.3 How Analysis is taught -- 3.4 Studying proofs -- 3.5 Self-explanation in mathematics -- 3.6 Proofs and proving -- 4 Learning Analysis -- 4.1 The Analysis experience -- 4.2 Keeping up -- 4.3 Avoiding time-wasting -- 4.4 Getting your questions answered -- 4.5 Adjusting your strategy -- Part 2 Concepts in Analysis -- 5 Sequences -- 5.1 What is a sequence? -- 5.2 Representing sequences -- 5.3 Sequence properties: monotonicity -- 5.4 Sequence properties: boundedness and convergence -- 5.5 Convergence: intuition first -- 5.6 Convergence: definition first -- 5.7 Things to remember about convergence -- 5.8 Proving that a sequence converges -- 5.9 Convergence and other properties -- 5.10 Combining convergent sequences -- 5.11 Sequences that tend to infinity -- 5.12 Looking ahead -- 6 Series -- 6.1 What is a series? -- 6.2 Series notation -- 6.3 Partial sums and convergence -- 6.4 Geometric series again -- 6.5 A surprising example -- 6.6 Tests for convergence -- 6.7 Alternating series -- 6.8 A really surprising example -- 6.9 Power series and functions -- 6.10 Radius of convergence -- 6.11 Taylor series -- 6.12 Looking ahead -- 7 Continuity -- 7.1 What is continuity? -- 7.2 Function examples and specifications -- 7.3 More interesting function examples -- 7.4 Continuity: intuition first. and 7.5 Continuity: definition first -- 7.6 Variants of the definition -- 7.7 Proving that a function is continuous -- 7.8 Combining continuous functions -- 7.9 Further continuity theorems -- 7.10 Limits and discontinuities -- 7.11 Looking ahead -- 8 Differentiability -- 8.1 What is differentiability? -- 8.2 Some common misconceptions -- 8.3 Differentiability: the definition -- 8.4 Applying the definition -- 8.5 Non-differentiability -- 8.6 Theorems involving differentiable functions -- 8.7 Taylor's Theorem -- 8.8 Looking ahead -- 9 Integrability -- 9.1 What is integrability? -- 9.2 Areas and antiderivatives -- 9.3 Approximating areas -- 9.4 Integrability definition -- 9.5 A non-integrable function -- 9.6 Riemann's condition -- 9.7 Theorems involving integrable functions -- 9.8 The Fundamental Theorem of Calculus -- 9.9 Looking ahead -- 10 The Real Numbers -- 10.1 Things you don't know about numbers -- 10.2 Decimal expansions and rational numbers -- 10.3 Rational and irrational numbers -- 10.4 Axioms for the real numbers -- 10.5 Completeness -- 10.6 Looking ahead -- Conclusion -- Bibliography -- Index.
Summary
Analysis is a core subject in most undergraduate mathematics degrees. It is elegant, clever and rewarding to learn, but it is hard. Even the best students find it challenging, and those who are unprepared often find it incomprehensible at first. This book aims to ensure that no student need be unprepared.
