### Amendments to Syllabus

In light of the COVID-19 Pandemic, all classes at NAU will be taught via remote instruction for the remainder of the Spring 2020 semester. Changes to the syllabus and class structure are outlined below. As we adjust to this new paradigm, we will need to adapt and improvise. Further changes might be necessary. Patience and compassion are vital.

• Starting Monday, March 23, I will use our shared Discord server to communicate. Please see your email for an invite code. I suggest you download the free Discord apps for your phone and computer. Contact me if you are having trouble using Discord.
• We may have some synchronous live classes (using Zoom), but for the most part we will take an asynchronous approach. This will maximize flexibility and minimize reliance on technology and bandwidth. Any live classes will be announced in advance and will be recorded for those that cannot make it. Please create a free Zoom account. I suggest you download the free Zoom apps for your phone and computer. Contact me if you are having trouble using Zoom.
• I will have several hours of virtual office hours (using Zoom). I will post my scheduled office hours soon. I will also be available via Discord (text and video).
• I will occasionally make videos to introduce new topics and to review solutions to a subset of the homework problems. I will post links to these videos in BbLearn and Discord.
• I plan to trim the content we cover to the absolute essentials. If we have time for extra stuff near the end of the semester, we will fill some stuff in. I have no doubt that we will cover a sufficient amount of material.
• Each of the remaining exams will be take-home exams. The usual rules apply. Due to the limited time we have left, you will likely have less time to work on the take-home exams than the previous one. I’ll take this into account when writing the exams. The release dates for the exams have yet to be determined.
• There will now be two types of homework assignments: Daily vs Weekly. Those of you that have had me before are somewhat familiar with this scheme, but my implementation will be different in this case.
• Daily Homework:
• Daily Homework is essentially the same as what we were previously calling “Homework”.
• For each assignment, you will need to capture your handwritten work digitally and then upload a PDF to BbLearn.
• The Daily Homework assignments will consist of a combination of working on new problems together with reflecting on a selection of student work from a previous assignment.
• Students are highly encouraged to discuss the homework on Discord. This includes reflecting on student work from previous problems. Engaging on Discord will have a positive impact on your Presentation and Participation grade.
• Daily Homework will be graded using a $\checkmark$-system: $\checkmark- = 1$, $\checkmark=2$, $\checkmark+=4$. Your grade will be determined by how much mathematics you produce. I’m looking for a good faith effort on completeness as opposed to doing everything perfectly.
• Weekly Homework:
• The purpose of the Weekly Homework will be to reflect on your work done in the Daily Homework and to ensure you understand how to do the problems on your second pass.
• Every few days, I will post solutions (likely using video) to a subset of the problems that were recently due. Your task will be to reflect on the work you previously did for the corresponding problems and to take notes on my proposed solutions.
• As with the Daily Homework, you will submit your work as a PDF to BbLearn.
• Each Weekly Homework assignment will be worth 4 points and your score will be based on the quality of the work that you submit.
• The remainder of your Presentation and Participation grade will be determined by your engagement with me and other students on Discord and in Zoom for office hours and/or live classes. With that being said, I want the engagement to be meaningful. I’m not interested in each person trying to reach a quota for number of posts. The material is sufficiently difficult that helping each other out just makes sense.
• The remaining content for this course is challenging and shifting to this new format will likely complicate things. However, if you engage and put in time and effort, everything will be okay. We are in this together and I’m here to assist.
• I know that you have other classes/commitments, so I’ll do my best to be reasonable with the workload.

### Course Info

Title: MAT 431: Introduction to Analysis
Semester: Spring 2020
Credits: 3
Section: 1
Time: 10:20-11:10AM
Location: AMB 147

### Instructor Info

Dana C. Ernst, PhD
AMB 176
9:00-10:00AM MWF, 9:00-11:00AM T
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. According to Laursen and Rasmussen (PDF), the Four Pillars of IBL are:

• Students engage deeply with coherent and meaningful mathematical tasks.
• Students collaboratively process mathematical ideas.
• Instructors inquire into student thinking.
• Instructors foster equity in their design and facilitation choices.

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.

### Problem Sequence

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. 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.

### Exams

There will be two midterm exams and a cumulative final exam. All exams will consist of an in-class part and may consist of a take-home part. The in-class portions of Exam 1 and Exam 2 will be taken individually out of class during a two-hour time block that you will schedule in advance. Exam 1 is tentatively scheduled for Friday, February 21 (week 6) and Exam 2 is scheduled for Friday, April 10 (week 12). We will not have class on the days that Exams 1 and 2 are given. The reason for the two-hour time block is to reduce pressure and eliminate time constraints. I will provide additional information about signing up for your two-hour time block as we get closer to Exam 1. Each of Exams 1 and 2 are worth 20% of your overall grade in the course. The final exam will be on Monday, May 4 at 10:00AM-12:00PM and is worth 20% of your overall grade. 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.

### Homework

Homework will usually be assigned each class meeting, and students are expected to complete each assignment before walking into the next class period. All assignments should be carefully, clearly, and cleanly written. Among other things, this means your work should include proper grammar, punctuation, and spelling. You should write a draft of a given solution before you write down the final argument, so do yourself a favor and get in the habit of differentiating your scratch work from your submitted assignment. The homework will generally consist of completing exercises and proving theorems from the problem sequence. 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 rubric given below. Not completing the self-assessment may impact the score on your homework.

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

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 problems that are due that day. At the end of each class session, you should submit the written work for all of your proposed solutions for problems that are due that day. On each homework assignment, I will provide grade of 0-2 based on the completeness of the assignment (not correctness). In addition, I will grade 1-2 problems (perhaps ones that were not presented in class), where each problem that is graded is worth either 2 or 4 points. If the problem is worth 4 points, it is subject to the following rubric. A scaled version of this rubric is used if the problem is worth 2 points.

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.

You are allowed (in fact, encouraged!) to modify your written work in light of presentations made in class; however, you are required to use the colored marker pens provided in class. This will allow me to differentiate the work done in class versus the work you completed before class. The grade you receive on an assignment will be determined by the work you completed prior to class. When annotating your work, keep in mind that I need to be able to clearly see what you wrote prior to class since this is the work that I am grading.

Please understand that the purpose of the homework assignments is to teach you to prove theorems in the context of real analysis. It is not expected that you started the class with this skill; hence, some low grades are to be expected. However, I expect that everyone will improve dramatically. Improvement over the course of the semester will be taken into consideration when assigning grades. Up to five times during the semester, you may resubmit a graded problem subject to the following constraints:

• A graded problem that received a score of 1, 2, or 3 is eligible to be resubmitted. Resubmissions on problems that received a 0 are not allowed.
• Resubmissions are due one week after the corresponding problem has been returned to the class.
• The grade on the resubmitted problem will replace the original score.
• Please write “Resubmission” on top of any problem that you are resubmitting and keep separate from any other problems that you are turning in.

You are allowed and encouraged to work together on homework. However, each student is expected to turn in their 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.

On each homework assignment, please write (i) your name, (ii) name of course, and (iii) assignment 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. Your overall homework grade will be worth 20% of your final grade.

### Presentations and Participation

The problems chosen for presentations will come from the Daily 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 their 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. 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.

90-100% Student receives M or E for participation. Student averages at least 1 presentation every week. 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 roughly 1.5 presentations every two weeks. Student occasionally volunteers to present. Most presentations receive M.
70-79% Student receives M for participation. Student averages roughly 1 presentation every two weeks. 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.

### 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. Extra credit points are only available to the first person to notify me of the error. They’re is a typo right here.

### Basis for Evaluation

Category Weight Notes
Homework 20% See above for requirements
Presentations & Participation 20% See above for requirements
Exam 1 20% In-class portion on Friday, February 21
Exam 2 20% In-class portion on Friday, April 10
Final Exam 20% Monday, May 4 at 10:00AM-12:00PM

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:

• Monday, January 20: MLK Day (no classes)
• Thursday, January 23: Last day to Add (without having to file a petition and pay a late add fee)
• Thursday, January 23: Last day to Drop/Delete (without class appearing on students’ transcripts)
• Monday, March 16-Friday, March 20: Spring break (no classes)
• Friday, March 23: Course withdrawal deadline
• Monday, May 4: Final Exam

### Getting Help

There are many resources available to get help. First, I recommend that you work on homework in small groups as much as possible and to come see me whenever necessary. 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! Lastly, you can always email me.

### 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
Website
928.523.6852
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