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.
- 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
Table of Contents
Chapter P1: Exploring Mathematical Statements
Chapter P2: Proving Mathematical Statements
Chapter P3: Preliminary Content
Chapter 1: Properties of R
Chapter 2: Accumulation Points and Closed Sets
Chapter 3: Open Sets and Open Covers
Chapter 4: Sequences and Convergence
Chapter 5: Subsequences and Cauchy Sequences
Chapter 6: Functions, Limits, and Continuity
Chapter 7: Connected Sets and the Intermediate Value Theorem
Chapter 8: Compact Sets
Chapter 9: Uniform Continuity
Chapter 10: Introduction to the Derivative
Chapter 11: The Extreme and Mean Value Theorems
Chapter 12: The Definite Integral: Part I
Chapter 13: The Definite Integral: Part II
Chapter 14: The Fundamental Theorem(s) of Calculus
Chapter 15: Series
Chapter E1: Function Approximation
Chapter E2: Power Series
Chapter E3: Sequences and Series of Functions
Chapter E4: Metric Spaces
Chapter E5: Iterated Functions and Fixed Point Theorems
John Ross is an Assistant Professor of Mathematics at Southwestern University. He earned his Ph.D. and M.A. in Mathematics from Johns Hopkins University, and his B.A. in Mathematics from St. Mary's College of Maryland. His research is in geometric analysis, answering questions about manifolds that arise under curvature flows. He enjoys overseeing undergraduate research, teaching in an inquiry-based format, biking to work, and hiking in Central Texas.
Kendall Richards is a Professor of Mathematics at Southwestern University. He earned his B.S. and M.A. in Mathematics from Eastern New Mexico University and his Ph.D. in Mathematics from Texas Tech University. He is inspired by working with students and the process of learning. His research pursuits have included questions involving special functions, inequalities, and complex analysis. He also enjoys long walks and a strong cup of coffee.
"Analysis has the potential to be one of the most enjoyable and challenging courses in the undergraduate curriculum. Taught poorly, it can devastate a student. Taught well, it can launch a student into a life-long love of theoretical mathematics. Introductory Analysis: An Inquiry Approach makes the latter both possible and probable. The authors strike the delicate balance between breadth and depth. They cover sufficiently many topics to satisfy any instructor, while delivering the material in such a way as to be amenable to an active- or inquiry-based learning pedagogy. The book could serve as either one or two semesters of undergraduate analysis and would be equally appropriate at either regional or research institutions. I commend the authors on this rich delivery and look forward to experimenting with the material myself."
—W. Ted Mahavier, Professor of Mathematics at Lamar University and Managing Editor for The Journal of Inquiry-Based Learning in Mathematics