Course Info

Title: MAT 431: Introduction to Analysis
Semester: Spring 2021
Credits: 3
Section: 1
Time: 8:00-8:50AM MWF
Location: AMB 225

Instructor Info

  Dana C. Ernst, PhD
  AMB 176
  MWF 1:30-2:30PM, T 1:00-2:00PM via Zoom
  dana.ernst@nau.edu
  928.523.6852
  danaernst.com

Prerequisites

MAT 320 with a grade of C or better.

Learning Outcomes

Upon completion of the course, students will be able to:

  1. Express an understanding of and apply basic properties and concepts of elementary analysis.
  2. Find limits, accumulation points, derivatives, and other objects and features of analysis.
  3. Identify and verify properties of given objects, e.g. convergence of a sequence; topological properties of a set such as openness and compactness; continuity; differentiability; integrability.
  4. Construct proofs in analysis.
  5. Determine truth or falsity of statements regarding analysis.
  6. Devise conjectures and construct examples within the area of analysis.
  7. Express work and results in coherent form using correct language and mathematics.

What is This Course All About?

This course introduces basic concepts and methods of analysis. The course focuses on the theory of the real number system and calculus of functions of a real variable. The content will include:

  • The Real Number System: axioms; supremum and infimum.
  • Topology of the real number system including completeness, compactness.
  • Sequences and Convergence, including the algebra of limits.
  • Limits of Functions, including the algebra of limits.
  • Continuity, including the algebra of continuous functions, continuity of compositions, and uniform continuity.
  • Differentiation, including the algebra of derivatives, chain rule, Mean Value Theorem, Inverse Function Theorem, applications to behavior of functions, Taylor’s Theorem and L’Hospital’s Rule.
  • Riemann integration, including linearity and order properties, integrability of continuous functions, Riemann sums, the Fundamental Theorem of Calculus.

We will take an axiomatic approach (definition, theorem, and proof) to the subject, but along the way, you will develop intuition about the objects of real analysis and pick up more proof-writing skills. The emphasis of this course is on your ability to read, understand, and communicate mathematics in the context of real analysis.

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.

Goals

Aside from the obvious goal of wanting you to learn how to write rigorous mathematical proofs, one of my principle ambitions is to make you independent of me. Nothing else that I teach you will be half so valuable or powerful as the ability to reach conclusions by reasoning logically from first principles and being able to justify those conclusions in clear, persuasive language (either oral or written). Furthermore, I want you to experience the unmistakable feeling that comes when one really understands something thoroughly. Much “classroom knowledge” is fairly superficial, and students often find it hard to judge their own level of understanding. For many of us, the only way we know whether we are “getting it” comes from the grade we make on an exam. I want you to become less reliant on such externals. When you can distinguish between really knowing something and merely knowing about something, you will be on your way to becoming an independent learner. Lastly, it is my sincere hope that all of us (myself included) will improve our oral and written communications skills.

An ounce of practice is worth more than tons of preaching.

An Inquiry-Based Approach

This is not a lecture-oriented class or one in which mimicking prefabricated examples will lead you to success. You will be expected to work actively to construct your own understanding of the topics at hand with the readily available help of me and your classmates. Many of the concepts you learn and problems you work on will be new to you and ask you to stretch your thinking. You will experience frustration and failure before you experience understanding. This is part of the normal learning process. If you are doing things well, you should be confused at different points in the semester. The material is too rich for a human being to completely understand it immediately. Your viability as a professional in the modern workforce depends on your ability to embrace this learning process and make it work for you.

In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL). Loosely speaking, IBL is a student-centered method of teaching mathematics that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Rather than showing facts or a clear, smooth path to a solution, the instructor guides and mentors students via well-crafted problems through an adventure in mathematical discovery. Effective IBL courses encourage deep engagement in rich mathematical activities and provide opportunities to collaborate with peers (either through class presentations or group-oriented work). If you want to learn more about IBL, read my blog post titled What the Heck is IBL?.

Don’t fear failure. Not failure, but low aim, is the crime. In great attempts it is glorious even to fail.

Much of the course will be devoted to students presenting their proposed solutions/proofs on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the word “produce” because I believe that the best way to learn mathematics is by doing mathematics. Someone cannot master a musical instrument or a martial art by simply watching, and in a similar fashion, you cannot master mathematics by simply watching; you must do mathematics!

In any act of creation, there must be room for experimentation, and thus allowance for mistakes, even failure. A key goal of our community is that we support each other—sharpening each other’s thinking but also bolstering each other’s confidence—so that we can make failure a productive experience. Mistakes are inevitable, and they should not be an obstacle to further progress. It’s normal to struggle and be confused as you work through new material. Accepting that means you can keep working even while feeling stuck, until you overcome and reach even greater accomplishments.

You will become clever through your mistakes.

Furthermore, it is important to understand that solving genuine problems is difficult and takes time. You shouldn’t expect to complete each problem in 10 minutes or less. Sometimes, you might have to stare at the problem for an hour before even understanding how to get started. In fact, solving difficult problems can be a lot like the clip from the Big Bang Theory located here.

In this course, everyone will be required to

  • read and interact with course notes and textbook on your own;
  • write up quality solutions/proofs to assigned problems;
  • present solutions/proofs on the board to the rest of the class;
  • participate in discussions centered around a student’s presented solution/proof;
  • call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar at first glance.

As the semester progresses, it should become clear to you what the expectations are.

Tell me and I forget, teach me and I may remember, involve me and I learn.

Course Structure

Class meetings will consist of discussion of problems, student-led presentations, and group work focused on problems selected by the instructor. A typical class session may include:

  • Informal student presentations of progress on previously assigned homework problems;
  • Summary of major steps and techniques of the solution of a finished problem;
  • Exploration of alternative approaches, possible generalizations, consequences, special cases, converse;
  • Discussion of relationships to previously assigned or solved problems;
  • Assignment of new problems;
  • Explanation of unfamiliar mathematical concepts as needed.

The impediment to action advances action. What stands in the way becomes the way.

Rights of the Learner

As a student in this class, you have the right:

  1. to be confused,
  2. to make a mistake and to revise your thinking,
  3. to speak, listen, and be heard, and
  4. to enjoy doing mathematics.

You may encounter many defeats, but you must not be defeated.

Commitment to the Learning Community

In our classroom, diversity and individual differences are respected, appreciated, and recognized as a source of strength. Students in this class are encouraged and expected to speak up and participate during class and to carefully and respectfully listen to each other. Every member of this class must show respect for every other member of this class. Any attitudes or actions that are destructive to the sense of community that we strive to create are not welcome and will not be tolerated. In summary: Be good to each other. I would appreciate private responses to the following question: Are there aspects of your identity that you would like me to attend to when forming groups, and if so, how?

Students are also expected to minimize distracting behaviors. In particular, every attempt should be made to arrive to class on time. If you must arrive late or leave early, please do not disrupt class. Please turn off the ringer on your cell phone. I do not have a strict policy on the use of laptops, tablets, and cell phones. You are expected to be paying attention and engaging in class discussions. If your cell phone, etc. is interfering with your ability (or that of another student) to do this, then put it away, or I will ask you to put it away.

Textbook

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 if necessary. The problem sequence is available here.

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?

Rules of the Game

Reviewing material from previous courses and looking up definitions and theorems you may have forgotten is fair game. However, when it comes to completing assignments for this course, you should not look to resources outside the context of this course for help. That is, you should not be consulting the web, other texts, other faculty, or students outside of our course in an attempt to find solutions to the problems you are assigned. This includes Chegg and Course Hero. On the other hand, you may use each other, the textbook, me, and your own intuition. If you feel you need additional resources, please come talk to me and we will come up with an appropriate plan of action. Please read NAU’s Academic Integrity Policy.

Homework

You are allowed and encouraged to work together on homework. However, each student is expected to turn in his or her own work. In general, late homework will not be accepted. However, you are allowed to turn in up to three late homework assignments with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. When doing your homework, I encourage you to consult the Elements of Style for Proofs. Your overall homework grade will be worth 20% of your final grade.

On each homework assignment, please write (i) your name, (ii) name of course, and (iii) Homework number. You can find the list of assignments on the homework page. I reserve the right to modify the homework assignments as I see necessary.

Homework will be assigned almost every class meeting, and students are expected to complete (or try their best to complete) each assignment before walking into the next class period. Homework will generally consist of completing exercises and proving theorems from the textbook. In addition to completing the assigned problems, I also want you to assess your own work. Next to each problem, I want you to write down a score between 0 and 4 that represents your perception of the validity and quality of your proposed solution/proof. Consider using the following rubric.

Grade Criteria
4 This is correct and well-written mathematics!
3 This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing.
2 There is some good intuition here, but there is at least one serious flaw.
1 I don't understand this, but I see that you have worked on it; come see me!
0 I believe that you have not worked on this problem enough or you didn't submit any work.

Not completing the self-assessment step may impact the score on your homework. On the day that a homework assignment is due, the majority of the class period will be devoted to students presenting some subset (maybe all) of the proofs/solutions that are due that day. With a few exceptions, homework assignment will consist of two parts.

Part 1: This portion of the assignment is named “Homework X” in BbLearn, where “X” is the number of the assignment. Prior to the start of class, you will need to capture your handwritten work digitally and then upload a PDF to BbLearn. There are many free smartphone apps for doing this. I use TurboScan on my iPhone. Submitting your work prior to class allows me to see what you accomplished outside of class. Part 1 will be assessed using the following rubric.

Grade Criteria
4 Sufficient effort was put into nearly all the problems.
2 Some problems were omitted and/or sufficient effort was not exhibited.
1 Many problems omitted and/or minimal effort exhibited.
0 Assignment was not turned in.

Part 2: This portion of the assignment is named “Notes X.Y” in BbLearn, where “X.Y” is the date of class. During class, we will discuss most of the problems that are due that day. While we are discussing them, you should either annotate your work and/or take notes on separate paper. It’s expected that most of the work you did prior to class will need to be refined. It is your responsibility to process this in some way. Annotating your work or taking notes will increase the chances that you are processing the work in a meaningful way. If you choose to annotate your work, please use a different color than what you originally used to complete your assignment. After class, you will need to capture your annotations/notes digitally and then upload a PDF to BbLearn. Part 2 will be assessed using the following rubric.

Grade Criteria
2 Student annotated their work in a meaningful way and/or took sufficient notes.
1 Student annotated their work and/or took notes but effort and/or attention to detail was insufficient.
0 Student did not submit this portion of the assignment.

Presentations and Participation

General Comments

The problems chosen for presentations will come from the homework assignments. Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken seriously since they spearhead the work of the class. Here are some of my expectations for the presenter:

  • The purpose of class presentations is not to prove to me that the presenter has done the problem. It is to make the ideas of the solution clear to the other students.
  • Presenters should explain their reasoning as they go along, not simply write everything down and then turn to explain.
  • Fellow students are allowed to ask questions at any point and it is the responsibility of the person making the presentation to answer those questions to the best of his or her ability.
  • Since the presentation is directed at the students, the presenter should frequently make eye-contact with the students in order to address questions when they arise and also be able to see how well the other students are following the presentation.

Alone we can do so little; together we can do so much.

Assessing Presentations

Presentations will be assessed using the following criteria.

Specification Criteria
U Unsatisfactory. Minimal progress was made that included relevant information or the student was unprepared.
I In progress. The student made an honest attempt at the problem but recognized a flaw that prevented them from being able to complete the problem during the presentation. Alternatively, the student reported on their current progress on a problem and attempted to convey where or why they are currently "stuck".
M Meets Expectations. The student demonstrated an understanding of the problem and presented the key ideas. Perhaps some details were omitted or interesting mistakes were made. The presentation led to fruitful class discussion.
E Exceeds Expectations. The presentation was flawless and the student demonstrated keen insight into the problem. The presentation led to fruitful class discussion.

You should aim to avoid unsatisfactory (U) presentations. An in progress (I) presentation should not be viewed as a bad thing as each of us will occasionally get stuck. However, you should strive for the majority of your presentations to meet (M) or exceed (E) expectations. Most presentations will meet expectations (M) while presentations that exceed expectations (E) will be rare. You should not let the rubric deter you from presenting if you have an idea about a solution/proof that you’d like to present but are worried that your solution/proof is incomplete or you are not confident your solution/proof is correct. You will be rewarded for being courageous and sharing your creative ideas! In my view, an interestingly wrong solution or proof makes for the best presentation since it generates the best discussion. This is really what we are after. On the other hand, you should not come to the board to present unless you have spent time thinking about the problem and have something meaningful to contribute. I will provide a progress report concerning each student’s presentation history after each of the midterm exams.

I alone cannot change the world, but I can cast a stone across the waters to create many ripples.

Selecting Presenters

In general, I will curate the list of student presenters each class meeting. A presenter is a student that either volunteered (V) or was chosen (C) by me. Volunteering is encouraged, but being chosen without volunteering isn’t bad. If more than one student volunteers for a specific problem, the student with the fewest number of presentations has priority. I reserve the right to decline your offer to present. This may happen if you are volunteering too often (and hence removing another student’s opportunity to present) or if I know in advance that another student’s presentation will lead to a fruitful discussion.

If you are chosen to present but would prefer not to present that particular problem, you can either negotiate presenting a different problem or take a pass (P). You may elect to pass at most two times during the semester, after which a presentation will be deemed unsatisfactory (U). By default, if you have an unexcused absence on a day when you have been chosen to present, then your presentation will be recorded as a pass (P) unless you have already exhausted your two passes, in which case the presentation will be recorded as unsatisfactory (U).

Specification Criteria
V Volunteered. Student volunteered during class or in advance to present.
C Chosen. Student was selected by the instructor and agreed to present.
P Pass. Student was selected to present, but asked to take a pass. Allowed at most two.

In summary, for each student presentation, I will record one of V, C, or P. In the case of V or C, I will also record one of U, I, M, or E based on the rubric given above. The most common pair will likely be CM (i.e., student was chosen to present and presentation met expectations).

Participation

You are expected to respectfully participate and contribute to class discussions. This includes asking relevant and meaningful questions to both the instructor and your peers. Moreover, you are expected to be engaged and respectful during another student’s presentation. Posting questions and/or responses in our Discord server counts as class participation, but posting in Discord is not required. Your class participation will be assessed as follows.

Specification Criteria
U Unsatisfactory. Student was often disengaged or disrespectful. Alternatively, the student regularly missed class.
M Meets Expectations. Student was consistently respectful, engaged, and contributed to meaningful class discussions. In addition, the student regularly attends class.
E Exceeds Expectations. Student's presence in the classroom truly enhances the learning environment.

Determining Presentation and Participation Grades

Your Presentation and Participation grade is determined by your frequency and ability to foster productive class discussions through presentations and audience participation. The greatest determining factor in your Presentation and Participation grade is your willingness to present often. You should aim to present at least twice prior to each midterm exam. The table below provides a summary of how your Presentation and Participation grade will be determined.

Grade Range Criteria
90-100% Student receives M or E for participation. Student averages at least 3 presentations prior to each of the exams. Student often volunteers to present and some of these problems are challenging. Most presentations receive M or E.
80-89% Student receives M or E for participation. Student averages at least 2 presentations prior to each of the exams. Student occasionally volunteers to present. Most presentations receive M.
70-79% Student receives M for participation. Student averages less than 2 presentations prior to each of the exams. Student rarely volunteers to present and actively avoids presenting challenging problems. Some presentations receive U.
60-69% Student receives U for participation. Student rarely presents and actively avoids presenting challenging problems. Some presentations receive U.
Below 60% Student receives U for participation. Student rarely or never presents and has completely disengaged from the class community.

I anticipate that most students will fall in the 80-89% range. Your Presentation and Participation grade is worth 20% of your overall grade.

I must not fear.
Fear is the mind-killer.
Fear is the little-death that brings total obliteration.
I will face my fear.
I will permit it to pass over me and through me.
And when it has gone past I will turn the inner eye to see its path.
Where the fear has gone there will be nothing.
Only I will remain.

Exams

There will be two midterm exams and a cumulative final exam. Each exam will consist of an “in-class” and possibly a take-home portion. Exam 1 and Exam 2 are tentatively scheduled for Friday, February 19 (week 6) and Friday, April 2 (week 12), respectively. Each of Exam 1 and Exam 2 will be worth 20% of your overall grade. The final exam will be on Monday, April 26 at 7:30-9:30AM and is worth 20% of your overall grade. If any of the exams have a take-home portion, you will have a few days to complete it. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time.

I write one page of masterpiece to ninety-one pages of shit.

Attendance

Regular attendance is expected and is vital to success in this course, but you will not explicitly be graded on attendance. Yet, repeated absences may impact your participation grade (see above). Students can find more information about NAU’s attendance policy on the Academic Policies page.

Extra Credit

The only thing I will award extra credit for is finding typos on course materials (e.g., textbook, exams, syllabus, webpage). This includes broken links on the webpage. However, it does not include the placement of commas and such. If you find a typo, I will add one percentage point to your next exam. You can earn at most two percentage points per exam and at most five percentage points over the course of the semester. They’re is a typo right here.

Basis for Evaluation

In summary, your final grade will be determined by your scores in the following categories.

Category Weight Notes
Homework 20% See above for requirements
Presentations & Participation 20% See above for requirements
Exam 1 20% In-class portion on February 19, possible take-home portion
Exam 2 20% In-class portion on April 2, possible take-home portion
Final Exam 20% April 26 at 7:30-9:30AM

It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.

Department and University Policies

You are responsible for knowing and following the Department of Mathematics and Statistics Policies (PDF) and other University policies listed here (PDF). More policies can be found in other university documents, especially the NAU Student Handbook (see appendices) and the website of the Student Outreach and Support (SOS).

As per Department Policy, cell phones, MP3 players and portable electronic communication devices, including but not limited to smart phones, cameras and recording devices, must be turned off and inaccessible during in-class tests. Any violation of this policy will be treated as academic dishonesty.

Important Dates

Here are some important dates:

  • January 18: Martin Luther King Day (no classes)
  • January 20: Last day to drop a course without a “W”
  • March 14: Last day to drop a course without a petition
  • April 26: Final Exam (7:30-9:30AM)

Getting Help

There are many resources available to get help. First, you are allowed and encouraged to work together on homework. However, each student is expected to turn in his or her own work. You are strongly encouraged to ask questions in our Discord discussion group, as I (and hopefully other members of the class) will post comments there for all to benefit from. You are also encouraged to stop by during my office hours and you can always email me. I am always happy to help you. If my office hours don’t work for you, then we can probably find another time to meet. It is your responsibility to be aware of how well you understand the material. Don’t wait until it is too late if you need help. Ask questions!

Changes to the Syllabus

Any changes to this syllabus made during the term will be properly communicated to the class.

If you want to sharpen a sword, you have to remove a little metal.

–>


Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
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Current Courses

  MAT 431: Intro to Analysis
  MAT 526: Combinatorics

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