Introductory Analysis: An Inquiry Approach

Description

Introductory Analysis: An Inquiry Approach aims to provide a self-contained, inquiry-oriented approach to undergraduate-level real analysis. The presentation of the material in the book is intended to be "inquiry-oriented'" in that as each major topic is discussed, details of the proofs are left to the student in a way that encourages an active approach to learning. The book is "self-contained" in two major ways: it includes scaffolding (i.e., brief guiding prompts marked as Key Steps in the Proof) for many of the theorems. Second, it includes preliminary material that introduces students to the fundamental framework of logical reasoning and proof-writing techniques. Students will be able to use the guiding prompts (and refer to the preliminary work) to develop their proof-writing skills. Features Structured in such a way that approximately one week of class can be devoted to each chapter Suitable as a primary text for undergraduates, or as a supplementary text for some postgraduate courses Strikes a unique balance between enquiry-based learning and more traditional approaches to teaching Cover Half Title Title Page Copyright Page Contents Preface Prerequisites P1. Exploring Mathematical Statements P1.1 What is a mathematical statement? P1.2 Basic set theory P1.3 Quantifiers, both existential and universal P1.4 Implication: the heart of a “provable” mathematical statement P1.5 Negations P1.6 Statements related to implication P2. Proving Mathematical Statements P2.1 Using definitions P2.2 Proving a basic statement with an existential quantifier P2.3 Proving a basic statement with a universal quantifier P2.4 Proving an implication directly P2.5 Proof by contrapositive P2.6 Proof involving cases P2.7 Proof by contradiction P2.8 Proof by induction P2.9 Proving that one of two (or one of several) conclusions is true P3. Preliminary Content P3.1 Relations and equivalence P3.2 Functions P3.3 Inequalities and epsilons Main Content 1. Properties of R 1.1 Preliminary work 1.2 Main Theorems 1.3 Follow-up work 2. Accumulation Points and Closed Sets 2.1 Preliminary work 2.2 Main Theorems 2.3 Follow-up work 3. Open Sets and Open Covers 3.1 Preliminary work 3.2 Main Theorems 3.3 Follow-up work 4. Sequences and Convergence 4.1 Preliminary work 4.2 Main Theorems 4.3 Follow-up work 5. Subsequences and Cauchy Sequences 5.1 Preliminary Work 5.2 Main Theorems 5.3 Follow-up Work 6. Functions, Limits, and Continuity 6.1 Preliminary Work 6.2 Main Theorems 6.3 Follow-up Work 7. Connected Sets and the Intermediate Value Theorem 7.1 Preliminary Work 7.2 Main Theorems 7.3 Follow-up Work 8. Compact Sets 8.1 Preliminary Work 8.2 Main Theorems 8.3 Follow-up Work 9. Uniform Continuity 9.1 Preliminary Work 9.2 Main Theorems 9.3 Follow-up Work 10. Introduction to the Derivative 10.1 Preliminary Work 10.2 Main Theorems 10.3 Follow-up Work 11. The Extreme and Mean Value Theorems 11.1 Preliminary Work 11.2 Main Theorems 11.3 Follow-up Work 12. The Definite Integral: Part I 12.1 Preliminary Work 12.2 Main Theorems 12.3 Follow-up Work 13. The Definite Integral: Part II 13.1 Preliminary Work 13.2 Main Theorems 13.3 Follow-up Work 14. The Fundamental Theorem(s) of Calculus 14.1 Preliminary Work 14.2 Main Theorems 14.3 Follow-up Work 15. Series 15.1 Preliminary work 15.2 Main Theorems 15.3 Follow-up work Extended Explorations E1. Function Approximation E1.1 Taylor Polynomials and Taylor's Theorem E1.2 Interpolation E1.3 Divided Differences E1.4 A Hybrid Approach E2. Power Series E2.1 Introduction to Power Series E2.2 Differentiation of a Power Series E2.3 Taylor Series E3. Sequences and Series of Functions E3.1 Pointwise Convergence E3.2 Uniform Convergence and Uniformly Cauchy Sequences of Functions E3.3 Consequences of Uniform Convergence E4. Metric Spaces E4.1 What is a Metric Space? Examples E4.2 Metric Space Completeness E4.3 Metric Space Compactness E5. Iterated Functions and Fixed Point Theorems E5.1 Iterative Maps and Fixed Points E5.2 Contraction Mappings E5.3 Newton's Method Appendix A. Brief Summary of Ordered Field Properties Bibliography Index