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Full 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
Contents
Prerequisites. P1. Exploring Mathematical Statements. P2. Proving Mathematical Statements. P3. Preliminary Content. Main Content. 1. Properties of R. 2. Accumulation Points and Closed Sets. 3. Open Sets and Open Covers. 4. Sequences and Convergence. 5. Subsequences and Cauchy Sequences. 6. Functions, Limits, and Continuity. 7. Connected Sets and the Intermediate Value Theorem. 8. Compact Sets. 9. Uniform Continuity. 10. Introduction to the Derivative. 11. The Extreme and Mean Value Theorems. 12. The Definite Integral: Part I. 13. The Definite Integral: Part II. 14. The Fundamental Theorem(s) of Calculus. 15. Series. Extended Explorations. E1. Function Approximation. E2. Power Series. E3. Sequences and Series of Functions. E4. Metric Spaces. E5. Iterated Functions and Fixed Point Theorems
