An Inquiry-Based Approach to Real Analysis

We will not be using a traditional textbook this semester, but rather a problem sequence adopted for an inquiry-based learning (IBL) approach to real analysis. The problem sequence that we are using is an adaptation of the analysis notes by Karl-Dieter Crisman, which are a modified version of notes by W. Ted Mahavier. Both authors have been gracious enough to grant me access to the source of these notes, so that we can modify and tweak for our needs. You can find a link to our textbook below. The textbook is subject to change and improvements.

Table of Contents

  • Preface
  • Chapter 1: Introduction
    • 1.1 What is This Course All About?
    • 1.2 An Inquiry-Based Approach
    • 1.3 Structure of the Notes
    • 1.4 Rules of the Game
    • 1.5 Rights of the Learner
    • 1.6 Some Minimal Guidance
  • Chapter 2: Preliminaries
    • 2.1 Sets
    • 2.2 Induction and the Well-Ordering Principle
    • 2.3 Functions
  • Chapter 3: The Real Numbers
    • 3.1 The Field Axioms
    • 3.2 The Order Axioms
    • 3.3 Absolute Value and the Triangle Inequality
    • 3.4 Suprema, Infima, and the Completeness Axiom
    • 3.5 The Archimedean Property
  • Chapter 4: Standard Topology of the Real Line
    • 4.1 Open Sets
    • 4.2 Accumulation Points and Closed Sets
    • 4.3 Compact and Connected Sets
  • Chapter 5: Sequences
    • 5.1 Introduction to Sequences
    • 5.2 Properties of Convergent Sequences
    • 5.3 Monotone Convergence Theorem
    • 5.4 Subsequences and the Bolzano–Weierstrass Theorem
  • Chapter 6: Continuity
    • 6.1 Introduction to Continuity
    • 6.2 Additional Characterizations of Continuity
    • 6.3 Extreme Value Theorem
    • 6.4 Intermediate Value Theorem
    • 6.5 Uniform Continuity
  • Chapter 7: Limits
    • 7.1 Introduction to Limits
    • 7.2 Limit Laws
  • Chapter 8: Differentiation
    • 8.1 Introduction to Differentiation
    • 8.2 Derivative Rules
    • 8.3 Mean Value Theorem
  • Chapter 9: Integration
    • 9.1 Introduction to Integration
    • 9.2 Properties of Integrals
    • 9.3 Fundamental Theorem of Calculus
  • Appendix A: Elements of Style for Proofs
  • Appendix B: Fancy Mathematical Terms
  • Appendix C: Definitions in Mathematics

I will not be covering every detail of the notes and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the content of the notes whenever necessary by asking questions. Here’s one of my favorite quotes about reading mathematics.

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?


Below are links to the take-home portions of each exam. If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam.

Miscellaneous Materials

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 226: Discrete Math
  MAT 690: CGT

About This Site

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Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.