Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer and they contain proofs of results. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

The notes presented here are primarily based on James Kirkwood's An Introduction to Analysis, 2nd Edition. This is in print by both PWS Publishing Company (Boston, MA; 1995) and Waveland Press (Long Grove, IL; 2002). A third edition is out from the CRC Press (Boca Raton, FL; 2021). In the Preface to the 3rd edition, Kirkwood comments: "The most significant change in this edition is the addition of a section on the Cantor set and the Cantor function and a small amount of material on connectedness." The new section is "Section 8.4. The Cantor Set and Cantor Function" and this will be covered in Analysis 2. These notes are largely based on the second edition, but will also incorporate the little bit of new material from the third edition.

The catalog description for Analysis 1 (MATH 4217/5217) is: "Studies elements of point set topology, limits and continuity, differentiability, Taylor's theorem, approximation, Riemann integral." This is a little inaccurate. A more appropriate description is: "Introduces the real numbers as a complete ordered field, covers open sets, sequences of real numbers, continuous functions, differentiable functions, and the Riemann integral and its properties." The topics of Riemann-Stieltjes integration, sequences of functions, series of real numbers, series of functions (including Fourier series), and analytic functions are explored in Analysis 2 (MATH 4227/5227). Traditionally, Chapters 1-5 of Kirkwood's book were covered in Analysis 1, and the remaining chapters were covered in Analysis 2. As a result of decreased enrollment in upper-level math classes, starting in fall 2025 Analysis 1 was restructured into "Analysis 1+" to include topics from Chapters 6 and 8 (and diminished emphasis on some of the assumed background material of Chapter 1).

Introduction.

• Section 0-0. Introduction. Section 0-0 notes

1. The Real Number System.

• Section 1-1. Sets and Functions. Section 1-1 notes
• Supplement. Proofs of Theorems in Section 1-1. Beamer file of Section 1-1 proofs
• Supplement. Printout of the Proofs of Theorems in Section 1-1. Print file of Section 1-1 proofs
• Section 1-2. Properties of the Real Numbers as an Ordered Field. Section 1-2 notes
• Supplement. Proofs of Theorems in Section 1-2. Beamer file of Section 1-2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 1-2. Print file of Section 1-2 proofs
• Section 1-3. The Completeness Axiom. Section 1-3 notes (This section includes an introduction to the cardinality of a set, a brief biography of Georg Cantor, and some history of the Archimedean Principle.)
• Supplement. Proofs of Theorems in Section 1-3. Beamer file of Section 1-3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 1-3. Print file of Section 1-3 proofs
• Study Guide for Chapter 1.

2. Sequences of Real Numbers.

• Section 2-1. Sequences of Real Numbers. Section 2-1 notes (This section includes a history of Cauchy sequences and completeness and discusses the conributions of Bolzano and Cauchy.)
• Supplement. Proofs of Theorems in Section 2-1. Beamer file of Section 2-1 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2-1. Print file of Section 2-1 proofs
• Section 2-2. Subsequences. Section 2-2 notes
• Supplement. Proofs of Theorems in Section 2-2. Beamer file of Section 2-2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2-2. Print file of Section 2-2 proofs
• Section 2-3. The Bolzano-Weierstrass Theorem. Section 2-3 notes (This section has a proof that a sequence of real numbers is Cauchy if and only if it is convergent and a brief biography of Weierstrass.)
• Supplement. Proofs of Theorems in Section 2-3. Beamer file of Section 2-3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2-3. Print file of Section 2-3 proofs
• Supplement. The Real Numbers are the Unique Complete Ordered Field. Reals are Unique notes (This supplement gives a brief discussion of Richard Dedekind and Dedekind cuts.)
• Supplement. Proofs of Theorems in this Supplement. Beamer file of Reals are Unique proofs
• Supplement. Printout of the Proofs of Theorems in this Supplement. Print file of Reals are unique proofs
• Study Guide 2.

3. Topology of the Real Numbers.

• Section 3-1. Topology of the Real Numbers. Section 3-1 notes (This section gives a definition of a "topology" on a set and brief biographies of Heine and Borel.)
• Supplement. Proofs of Theorems in Section 3-1. Beamer file of Section 3-1 proofs
• Supplement. Printout of the Proofs of Theorems in Section 3-1. Print file of Section 3-1 proofs
• Supplement: A Classification of Open Sets of Real Numbers. Classification of Open Sets notes
  • Supplement. Proofs of Theorems in this Supplement. Beamer file of Classification of Open Sets proofs
  • Supplement. Printout of the Proofs of Theorems in this Supplement. Print file of Classification of Open Sets proofs
• Study Guide 3

4. Continuous Functions.

• Section 4-1. Limits and Continuity. Section 4-1 notes (This section includes an example of a function continuous on the irrationals and discontinuous on the rationals; see Example 4.7.)
• Section 4-2. Monotone and Inverse Functions. Section 4-2 notes
• Study Guide 4.

5. Differentiation.

• Section 5-1. The Derivative of a Function. Section 5-1 notes
• Section 5-2. Some Mean Value Theorems. Section 5-2 notes
• Study Guide 5.

6. Integration (Partial).

• Section 6-1. The Riemann Integral. Section 6-1 notes (This section gives a statement of the Riemann-Lebesgue Theorem which states that a bounded function is Riemann integrable if and only if its set of discontinuities has measure zero. This section also defines the Cantor set. For a proof of the Riemann Lebesgue Theorem, see The Riemann-Lebesgue Theorem handout which is used in graduate level Real Analysis 1.)
• Section 6-2. Some Properties and Applications of the Riemann Integral. Section 6-2 notes (This section defines the natural log function and establishes its properties.)
• Study Guide.

8. Sequences and Series of Functions (Partial).

• Section 8-1. Sequences of Functions. Section 8-1 notes (This section includes a proof that the uniform limit of a sequence of Riemann integrable functions is itself Riemann integrable [Theorem 8-3].)
• Study Guide.

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