Here is a link to "Real Analysis in Reverse" by James Propp.

**Exam 4 will be take home. Details and ground rules are below.**

**All assignments must be turned in by Thursday, December 8, 5:30pm.**

**8/26** Homework 1 is posted below. Due in class 9/2.

**9/15** Exam 1 is Wednesday, September 21, in class.

**Overview** This is an introductory course to proof-based mathematics. We will be interested in understanding the rigorous foundations of single variable calculus, in particular, properties of the real numbers, limits, continuity, differentiation, and the Riemann integral.

**Office** My office is 446 in the Mathematical Sciences building.

**Office hours** Monday 3:00-4:30, Thursday 4:00-5:30, or by appointment.

**Text** The textbook for this class is Elementary Real Analysis: Second Edition (2008) by Thomson, Bruckner, and Bruckner.

The text is available for **FREE** as a pdf download (click here).

A paperback copy may be ordered from CreateSpace or Amazon for $33.95.

**Academic calendar** For ease of reference, here is a link to the academic calendar detailing all breaks, add/drop deadlines, etc.

**Accommodations** In this mathematics course accommodations are managed between the instructor, student, and DRC Testing Center. Students should see instructors outside class hours before or after class or during office hours to share your Accommodation Memorandum for the current semester and discuss your accommodations as soon as possible.

**Exams** There will be four exams, the first three in class and the last during the final exam period. (See below for schedule and further details.) One exam will be dropped. Please contact me as soon as possible when you become aware of any exam conflicts or if you have missed an exam.

**Homework** There will be weekly homework sets. (See below for details and due dates.) Students are encouraged to collaborate on homeworks as long as: 1) all collaborators are listed on the cover sheet of each student's assignment; 2) each student turns in their own, individual work. Rote copying of solutions from peers, or plagiarism of any kind, will not be tolerated.

**Homework formatting** Homeworks must be typed or neatly written on loose leaf paper (no fringes!). There should be no significant cross-outs, rewrites, scratchwork, scribbling, etc. Problems should be clearly indicated and pages must be securely fastened together. Homework should have a cover sheet attached with your name, course section (either 62 or 64), the homework set number, and the names of your collaborators, in that order.

**Late homework** Late homework will only be accepted if turned in to me in person during office hours. As part of the conditions of acceptance, you may be asked to present your solution of a problem of my choosing. No penalty will be assessed for assignments which are not excessively late (less than one week past due).

**Quizzes** There will be several quizzes throughout the semester. Quizzes will be announced at least one class period in advance. At least one quiz will be dropped, maybe more depending on the total number of quizzes.

**Grades** After drops, each item in a given category receives equal weighting. Your percentage of points earned in each category will be combined into a single score with weighting 60% on exams, 30% on homeworks, and 10% on quizzes. Grades will be assigned based on rank among all students in both sections. For marginal cases, there will be some discretionary leeway in final grade assignment to account for course participation/engagement or extraordinary effort.

**Introduction - What is Analysis?**

Preliminary remarks.

**Topic 1 - Background I**

Functions; Sets; Set operations; Basic notation

corresponding sections in the text: A.1, A.2.

**Topic 2 - Background II**

Proof; Proof techniques; Propositional logic

corresponding sections in the text: A.4-A.9

**Topic 3 - The Real Numbers I**

Algebra; Order

corresponding sections in the text: 1.1-1.4

**Topic 4 - The Real Numbers II**

Metric structure; Sup and inf; On completeness

corresponding sections in the text: 1.5-1.10

**Topic 5 - Sequences I**

corresponding sections in the text: 2.1, 2.2, 2.5-2.8

**Topic 6 - Sequences II**

corresponding sections in the text: 2.9, 2.11-2.13

**Topic 7 - Topology of the Reals I**

corresponding sections in the text: 4.1-4.3

**Topic 8 - Topology of the Reals II**

corresponding sections in the text: 4.5-4.6

**Topic 9 - Countability**

corresponding sections in the text: 2.3, 4.6

**Assignment 1, Due: Friday, September 2, in class ** Questions A.2.1-A.2.7, A.2.10, A.2.12, A.2.13.

In addition the following supplementary question. Draw a picture "proof" for DeMorgan's laws. Now prove DeMorgan's laws rigorously. Graded exercises.

**Assignment 2, Due: Friday, Sept. 9, in class** Questions A.5.2, A.8.2-A.8.4, A.8.6, A.8.8, A.9.1, 1.4.3, 1.6.11, 1.6.12.

In addition the following supplementary question: Show that $\sqrt{3}$ is not rational. Graded exercises.

**Assignment 3, Due: Friday, Sept. 16, in class** Questions 1.6.23, 1.7.1, 1.7.7, 1.9.6, 2.2.8, 2.2.9. Graded exercises.

**Assignment 4, Due: Friday, Sept. 23, in class** Questions 2.4.2, 2.4.5, 2.4.14, 2.4.16, 2.5.6, 2.7.5, 2.7.7, 2.8.9, 2.11.17, 2.11.28. Graded exercises.

**Assignment 5, Due: Friday, Sept. 30, in class** Questions 2.12.6, 2.14.1, 2.14.2, 4.2.16, 4.4.3, 4.4.8, 4.4.10. Graded exercises.

**Assignment 6, Due: Friday, Oct. 7, in class** Questions 4.7.4, 4.7.5, 4.7.6, 4.7.7, 4.7.8. Graded exercises.

**Assignment 7, Due: Friday, Oct. 14, in class** $\emptyset$ Graded exercises.

**Assignment 8, Due: Friday, Oct. 21, in class** 5.2.18, 5.2.37, 5.4.13, 5.6.8, 5.6.9, 5.8.7, 5.9.14, 5.10.1, 5.10.12, 5.10.13. Graded exercises.

**Assignment 9, Due: Friday, Oct. 28, in class** 7.2.6, 7.2.12, 7.2.16, 7.5.3, 7.6.3. Graded exercises.

**Assignment 10, Due: Friday, Nov. 11, in class** 7.9.1, 7.10.5, 7.10.9, 7.13.3. Turn in exercises 7.13.4, 7.13.5, 7.13.7, 7.13.13 separately for up to 3 points bonus. Graded exercises.

**Assignment 11, Due: Friday, Nov. 19, in class** 3.2.17, 3.3.1, 3.3.2, 3.4.1, 3.5.2, 3.5.5, 3.6.2, 3.6.15, 3.6.16, 3.6.24. Graded exercises.

**Assignment 12, Due: Friday, Dec. 2, in class** 8.3.6, 8.5.4, 8.5.5, 8.5.6, 8.5.10, 8.5.11, 8.6.4, 8.6.5, 8.7.3, 8.10.2. Graded exercises.

**Quiz 1, Friday, Sept. 2** Solution.

**Quiz 2, Wednesday, Sept. 14** Solution.

**Quiz 3, Wednesday, Oct. 5** Solution.

**Quiz 4, Monday, Nov. 14** Solution.

**Quiz 5, Wednesday, Nov. 30**

**Wednesday, September 21** sample solutions.

**Wednesday, October 19** sample solutions.

**Topics 2.12, 2.13, 4.1-4.4, 4.5.1, 4.5.4, 5.1, 5.2, 5.4-5.8, 5.9.1, 5.9.2, and related lectures and homeworks**

**Friday, November 18** sample solutions

**Topics 7.1-7.7, 7.9, 7.10, 3.1, 3.2, 3.4, 3.5, 3.6.1-3.6.10, and related lectures and homeworks**

Due by **9:00pm, Wednesday, December 14.**

**Topics, Chapter 8 and related lectures and homeworks.**

**This exam may be taken at home. It should be available Thursday, December 8. Here are the ground rules.
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**Ready?** Here it is. Good luck!