324 Information

Mathematics 324: Real Analysis
Fall 2023
Introduction

Professor:        Jeff Johannes                           Section 1    MWF 11:30a-12:20p    South 328
Office:               South 326A
Telephone:     5403 (245-5403)
Office Hours:     Monday 5-6p in South 309, Tuesday 1-2p in South 309. Wednesday 12:30-1:20p in South 309, Thursday 8-9p in South 336, Friday 2:30-3:30p in South 309, and by appointment or visit.
Email Address: Johannes@Geneseo.edu
Web-page:         http://www.geneseo.edu/~johannes

Textbook

Introduction to Analysis, *any* edition, by Arthur Mattuck (link includes the first 3 chapters and errata)

Overview

What if you look back at Calculus I - II and focus all your attention on the proofs and reasons? This is what we do in Real Analysis. If you paid attention to all the reasons when you took the courses originally, this will all be familiar. If not, you will wish you had. We will start with "what is a real number" and end with the Fundamental Theorem(s) of Calculus.

Learning Outcomes

Upon successful completion of Math 324 - Real Analysis I, students will be able to:

Describe the real line as a complete, ordered field,
Determine the basic topological properties of subsets of the real numbers,
Use the definitions of convergence as they apply to sequences, series, and functions,
Determine the continuity, differentiability, and integrability of functions defined on subsets of the real line,
Apply the Mean Value Theorem and the Fundamental Theorem of Calculus to problems in the context of real analysis, and
Produce rigorous proofs of results that arise in the context of real analysis.
Write solutions to problems and proofs of theorems that meet rigorous standards based on content, organization and coherence, argument and support, and style and mechanics.

Grading

   Your grade in this course will be based upon your performance on homework, edits, examples and three exams. The weight assigned to each is designated below:
        Homework (8)    6% each
        In-class exams (2)    16% each
        Final exam (1)          20%

Reading

I have intentionally chosen a very readable text. In addition to planning time to do homework, please take time to carefully read the sections in the book. Notice use of the words “time" and “carefully". Read the sections slowly. Remember that the proofs are the most important part by far. If you read only the proofs you'll be … ok. If you read none of the proofs, it will be hopeless. Because the text is exceptionally accessible, we will structure class-time more as an interactive discussion of the reading than lecture. For each class day there is an assigned reading. Read and take notes on the section before coming to class. In addition to the reading, there are also questions in the text for each section. It would be good to practice with them. They will not be on the problem sets, but they are candidates for the exams. You are responsible for all sections in the textbook (up to and including 20.4), even if we do not explicitly discuss them in class. There is more detail in the text than we can possibly do in class. Bringing your questions will help guide our class conversation.

Problem Sets

    There will be eight problem sets due on indicated dates. The problems will be mostly proofs. You are encouraged to consult with me outside of class on any questions toward completing the homework. You are also encouraged to work together on homework assignments, but each must write up their own well-written solutions. A violation of this policy will result in a zero for the entire assignment and reporting to the Dean of Students for a violation of academic integrity. A good rule for this is it is encouraged to speak to each other about the problem, but you should not read each other's solutions. Each question will be counted in the following manner:
    0 missing or plagiarised question
    1 question copied
    2 partial question without significant progress - not eligible to rewrite
    3 significant progress, but incomplete question
    4 completed question with some solution
    5 completed question correctly and well-written
Each entire homework set will then be graded on a 90-80-70-60% (decile) scale. Late items will not be accepted.

Solutions and Plagiarism

There are plenty of places that one can find all kinds of solutions to problems in this class. Reading them and not referencing them in your work is plagiarism, and will be reported as an academic integrity violation. Reading them and referencing them is not quite plagiarism, but does undermine the intent of the problems. Therefore, if you reference solutions you will receive 0 points, but you will *not* be reported for an academic integrity. Simply - please do not read any solutions for problems in this class. Any work written, developed, or created, in whole or in part, by generative artificial intelligence (AI) is considered plagiarism and will not be tolerated. While the ever-changing developments with AI will find their place in our workforces and personal lives, in the realm of education and learning, this kind of technology does not help us achieve our educational goals. The use of AI prevents the opportunity to learn from our experiences and from each other, to play with our creative freedoms, to problem-solve, and to contribute our ideas in authentic ways. Geneseo is a place for learning, and this class is specifically a space for learning how to advance our thinking and professional practice. AI cannot do that learning for us.

Rewrites

You may resubmit any or all problems from your problem set no more than 3 class days (i.e. a class week) after the original due date. Include the original paper that I returned to you. On separate paper, include any rewrites that you would like graded. I will regrade any items that have earned at least 3 points originally. Please only resubmit problems that you want regraded. I will replace your original grade with the rewrite grade for each of the resubmitted problems. Addressing any comments I make on the original would seem to be a good idea. I am happy to work with you on rewrites as much as on the original.

Opening Meeting

Students will earn two extra points on the first problem set by visiting office hours during the first two weeks of classes, i.e. no later than 11 September.

Exams

The exams will consist of a few posers from the book (maybe with slight modifications, perhaps only a part instead of the full item). They will come from the "Questions" and "Exercises" and will not include "Problems".

Feedback

Occasionally you will be given anonymous feedback forms. Please use them to share any thoughts or concerns for how the course is running. Remember, the sooner you tell me your concerns, the more I can do about them. I have also created a web-site which accepts anonymous comments. If we have not yet discussed this in class, please encourage me to create a class code. This site may also be accessed via our course page on a link entitled anonymous feedback. Of course, you are always welcome to approach me outside of class to discuss these issues as well.

Social Psychology

Real Analysis is infamous for being difficult, right? Here are some different sides to that. There will be a significant amount of reading and writing. Avoiding either will be … detrimental. Here are the good sides: everything we do should be familiar. As I said before if you paid attention in Calculus I and II you will be in good shape. Since our class is so small, I can easily work with each of you individually in class and in office hours. The other side to that is … if you are not working it will be obvious. If you are not reading before each class and coming to office hours each week … this is unlikely to be successful. Like induction, there are two key steps - starting right, and continuing. I am ready to show my dedication to each of you, please return the courtesy.

Accessibility Accommodations

SUNY Geneseo is dedicated to providing an equitable and inclusive educational experience for all students. The Office of Accessibility (OAS) will coordinate reasonable accommodations for persons with disabilities to ensure equal access to academic programs, activities, and services at Geneseo. Students with approved accommodations may submit a semester request to renew their academic accommodations. Please visit the OAS website for information on the process for requesting academic accommodations. Contact the OAS by email, phone, or in-person: Office of Accessibility Services Erwin Hall 22 585-245-5112 access@geneseo.edu

Religious Holidays

    It is my policy to give students who miss class because of observance of religious holidays the opportunity to make up missed work. You are responsible for notifying me no later than September 11 of plans to observe the holiday.

Schedule (subject to change and clarification)

Date              Topic
August 28    Introduction + 1.1
        30 (extended version of Theorem 1.3)
September 1     2.1 - 4

          6       2.5 - 3.1
          8 3.1 - 3.6 PS1 (Chapters 1-2)

          11      3.7 - 4.4
       13    5.1 - 5.3
          15 5.4 - 6.1

          18    6.2 - 6.5
          20    7.1 - 7.3 PS2 (Chapters 3-4)
          22    7.4 - 7.7

          25    8.1 - 8.4
          27    9.1 - 9.4
          29    PS3 (Chapters 5-7)

October 2    review
          4         XM1 (Chapters 1-7)
          6

        11        9.5 - 10.2
          13      10.3 - 11.1

      16
          18    11.2 - 11.3        PS4 (Chapters 8-10)
          20      11.4 - 12.1

        23    12.2 - 13.3
25
27       PS5 (Chapters 11-12)

          30     13.4 - 14.1
November 1 14.2 - 15.2
          3

          6       PS6 (Chapters 13-14)
          8     review
          10 XM2 (Chapters 8-14)

          13     15.3 - 16.1
          15 16.2 - 17.2
          17 17.3 - 18.2

          20 18.3 - 19.3

        27      PS7 (Chapters 15-17)
          29     19.4 - 20.1
December 1 20.2 - 20.4

          4        20.5 - 20.6
          6
          8      review PS8 (Chapters 18-20)

          11      review

Friday, December 15 8:00 - 11:00a Final Exam (Half from 15-20, half from 1-14)