This is the main page for section **55380** of **Math 361K**.

This is a special section, which will be organized differently from a standard lecture-based course. The basic
principle is that *you* will *discover* the basic precepts of real analysis. So
in a typical class meeting you will be working among yourselves in groups to solve the day's
problems, presenting proofs on the blackboard and critiquing them, and both asking and answering
questions. My main role, and that of the teaching assistant, will be to provide overall
direction and guidance.

Broadly speaking, the goal is to cover: basic properties of the real numbers; sequences of real numbers, their
convergence, and completeness of the real numbers; limits of functions; continuity and uniform continuity
of real-valued functions in one variable;
differentiation of real-valued functions in one variable; integration of real-valued
functions in one variable. The treatment will be fully rigorous and proof-based.
The syllabus and schedule are necessarily *extremely tentative*, since in an IBL course a lot of what we do
will depend on the particular tastes and predilections of the class. Nevertheless, a rough plan follows
(to be updated throughout the semester):

- Meeting 1: introduction, basic ordered field properties of the real numbers
- Meetings 2-19: sequences and their convergence, completeness of the real numbers
- Meetings 20-25: limits of functions, continuity, properties of continuous functions (includes Intermediate Value Theorem)
- Meetings 26-27: differentiability and differentiation

I am Andrew (Andy) Neitzke. You can contact me at neitzke@math.utexas.edu.
My **office hours** are Tuesday 5:00-6:30p and Wednesday 4:00-5:00p, in **RLM 9.134**.

Our teaching assistant is Alice Mark. You can contact her at amark@math.utexas.edu.
Her **office hours** are Monday 5:30-6:30p, Thursday 5:00-6:00p and Friday 4:00-5:00p, in **RLM 13.150**.

Class meetings are **Tuesday and Thursday**, from **9:30a-11:00a**, in **RLM 6.126**.
Excluding Thanksgiving and the 1 midterm exam (see below) this gives a total of 27 meetings.

An important feature of IBL is that you should *not* consult a standard real
analysis textbook, nor any Internet resources; the idea is to struggle with the material
yourselves rather than just reading the answers somewhere.

A special text has therefore been developed for this course, which contains some of the basic definitions and helpful advice, but leaves most of the real work to you.

I will post this text in sections here as we go:

- Chapter 1. Introduction.
- Appendix A. Prerequisite Knowledge.
- Chapter 2. Preliminaries: Numbers and Functions.
- Chapter 3. Sequences.
- Appendix B. Cardinality, open and closed sets.
- Chapter 4. Continuous Functions.
- Chapter 5. Differentiability and Differentiation.
- Chapter 6. Integrability and Integration.

The course grade will be determined based on class participation (30%), homework (40%) and exams (30%). It will be assigned using the +/- system.

There will be one in-class midterm exam (on **Thu Oct 13**) and a final exam.
Here is a practice midterm. Here is the final exam.

Homework will be assigned during most class meetings, due at the following meeting. Most
of the homework problems will ask you to prove statements given in the text. Not
every problem will be graded, but we will grade as many as practicable.
*Working together on the homework is strongly encouraged, but you must write out your own solutions
individually,
and you must not use any resources other than your classmates, the text, or us.*

Here is a list of assignments so far:

- Assignment 1: Problems A.4, A.8, A.11, A.12, 2.6, 2.7, 2.8, 2.9, 2.10.
- Assignment 2, due 1 Sep (but to be collected 6 Sep): Problems 3.2, 3.4.
- Assignment 3, due 6 Sep: Problems 3.3, 3.5, 3.6, 3.7, 3.8.
- Assignment 4, due 8 Sep: Problems 3.9, 3.10, 3.11, 3.12, 3.13, 3.14.
- Assignment 5, due 13 Sep: Problems 3.15, 3.16, 3.17, 3.18, 3.19, 3.20.
- Assignment 6, due 15 Sep: Problems 3.21, 3.22, 3.23.
- Assignment 7, due 20 Sep: Problems 3.25, 3.26, 3.27, 3.28, 3.29, 3.30.
- Assignment 8, due 22 Sep: Problems 3.31, 3.32, 3.37.
- Assignment 9, due 27 Sep: Problems 3.38, 3.39, +3 older problems.
- Assignment 10, due 29 Sep: Problems 3.40, 3.41, 3.42.
- Assignment 11, due 4 Oct: Problems 3.43, 3.44, +3 older problems (suggested: 3.7, 3.17).
- Assignment 12, due 6 Oct: Problems 3.45, 3.46, +1 older problem.
- Assignment 13, due 11 Oct: Problems 3.47, 3.58, +3 older problems.
- No assignment due 13 Oct.
- Assignment 14, due 18 Oct: Problems 3.59, 3.60, 3.61, 3.62, 3.63.
- Assignment 15, due 20 Oct: Problems 3.64, 3.65, 3.66, 3.67.
- Assignment 16, due 25 Oct: Problems 3.68, 3.69, 3.70, 3.71, 3.72, 3.73.
- Assignment 17, due 27 Oct: Problems 3.74, 3.75, 3.76, 3.77.
- Assignment 18, due 1 Nov: six of Problems 4.1-4.10.
- Assignment 19, due 3 Nov: Problems 4.11, 4.14, +1 older problem; read 4.12, 4.13, 4.15.
- Assignment 20, due 8 Nov: Problems 4.16, 4.17, 4.18, 4.20 parts a and b, EITHER 4.21 OR 4.22, +1 older problem.
- Assignment 21, due 10 Nov: Problems 4.23, 4.24, 4.25.
- Assignment 22, due 15 Nov: Problems 4.26, 4.28, 4.29, 4.30, 4.31. Model solution for 4.31.
- Assignment 23, due 17 Nov: Problems 4.32, EITHER 4.33 or 4.34, 4.39; read 4.35-4.38.
- Assignment 24, due 22 Nov: Problems 4.41, 4.42, 4.43, 4.46, 4.47, 4.48, 4.49.
- Assignment 25, due 29 Nov: Problems 4.50, 4.51, and TWO OF 5.1-5.4.

A Web forum for the course is hosted on the department's Moodle server, here.

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.