Introductory Analysis : An Inquiry Approach.

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.

Introductory Analysis: An Inquiry Approach aims to provide a self-contained, inquiry-oriented approach to undergraduate-level real analysis. 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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.