MAT 401 Real Analysis

Undergraduate course, Mathematics Department, Utica College, 2019

Welcome to the home page of Dr. Xiao Xiao’s Real Analysis at Utica College. You can find all the informtion and documents for this course on this page. Please check this page frequently for announcements and assignments.

Important Dates

  • Add/Drop deadline: 9/6/19
  • Withdraw deadline: 11/11/19
  • Final exam: 12/20/19, 9:00 a.m. - 12:00 p.m.

Instructor Information

  • Instructor: Dr. Xiao Xiao
  • Email:
  • Office: White Hall 255
  • Office hour: MF 12:30 p.m. - 2:00 p.m., W 12:30 p.m. - 1:30 p.m. or by appointment.

An Important Course Policy

I pride myself on having a good environment for working and learning. It is very important to me that we all treat each other with care and respect, in equal measure. I know that I ask students to take risks in class almost every day, and this can be challenging for many. I ask that you help me keep our classroom a supportive place for each of the people in it. Each of us deserves the space to bring our full, authentic selves to class and be comfortable.

General Course Information and Policies

  • Course name: MAT 401 Real Analysis
  • Course credit hours: 3-credit
  • Course prerequisite: MAT 305, or permission of instructor
  • Class time and location: MWF 9:30 a.m. - 10:20 a.m. at Hubbard Hall 202
  • Textbook: We will not use a textbook, but rather a task-sequence adopted for inquiry-based learning. The task-sequence is written by David Clark and myself. You are expected to work out the tasks as the semester progresses.
  • Course description: Foundations of the real number system, functions and sequences, limits and continuity.
  • Program learning goals: In accordance to the learning goals of the Department of Mathematics of Utica College, MAT 401 will introduce and reinforce students’ ability of:
    • (PLG1) Reading and analyzing mathematical proofs.
    • (PLG2) Writing mathematical proofs.
    • (PLG5) Communicating mathematics in written form.
  • Course learning objectives: Upon successful completion of this course, students will be able to:
    • understand and prove theorems about basic topological properties of the real number line using basic definitions including open and closed sets, limit points, converging and diverging sequences and monotonic sequences.
    • understand and prove the theorems about the relationships between rational numbers and infinite decimal representations of numbers.
    • understand and prove all the theorems that lead to the theorem that the real number field is the smallest complete field extension of the rational numbers.
    • understand and prove theorems about basic properties of continuous functions.
  • Class organization: This course will likely be different from any other math course you have taken before. As an instructor, I will not be lecturing most of the time although I love lecturing very much. Scientific research shows that most people do not learn mathematics by listening, instead, they learn by doing it! I am sure you have said to yourself before “It looked so easy when the professor was doing it, but now I am confused when I have to do it by myself.” Why? Because the knowledge belongs to your professor and does not belong to you. You do not learn the knowledge simply by hearing it once or twice from somebody else. In order for you to have a more thorough understanding of the knowledge, we will incorporate ideas from an educational philosophy called the Moore method (after R. L. Moore). More precisely, we will use the modified Moore method, also known as inquiry-based learning. Most of the time during the class, students will be presenting proofs of theorems that they have produced by themselves, and not by other people or textbooks. A significant portion of your grade will be determined by how much mathematics you produce.
  • 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. On the other hand, you may use each other, the course notes, me, and your own intuition.
  • Regular attendance is mandatory 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.
  • MAT 401 is a writing intensive course. A writing intensive class has specific requirements on the style and the process of your writing. The first requirement is informal writing assignments which you will accomplish through your weekly journals and daily homework. The second requirement is a formal writing assignment that you will have the opportunity to revise after receiving feedback. Your work on the weekly homework and portfolio (which are based on your daily homework) include multiple submissions that support the writing process including an outline, rough draft, and revision of the mathematical writings describing your experiences.


  • Daily Homework

    Daily Homework will be assigned each class period, and students are expected to complete (or try their best 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 will almost always write a draft 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.

    Please use filler paper to write up Daily Homework, one problem per page. There is a very good reason for doing this, which will be explained later in the Portfolio section.

    Each Daily Homework will be submitted twice. The first time you will submit at Canvas before 12:00pm on the day before we meet in class. For example, a Daily Homework assigned on Monday will be discussed on Wednesday, so you should submit your draft by Tuesday 12:00pm at Canvas. I will review your draft and give suggestions and comments within 5 hours (namely before 5:00pm of the same day). You can read the comments and make necessary changes before you come to class on the next day. Your Daily Homework will be finished by hand and paper so the best way to do this is to take a picture of your writings and submit it as attachment on Canvas. Students must upload the draft in PDF format. You can use this website to convert JPG format to PDF format. In Windows, you can install CutePDF to convert any format to PDF format. In Mac, you can just choose “Print” and then on the bottom left corner choose “Save as PDF”. Please make sure that all pictures are properly oriented before uploading them to Canvas.

    Daily Homework are graded on the following rubrics (per problem):

    1Your solutions are basically correct or you have made some non-trivial progress towards the solutions of the problems but your solutions also have non-trivial gaps that need to be fill in.
    0You did not make any progress towards the solutions of the problems.

    Daily Homework will generally consist of proving theorems or propositions from the task-sequence. On the day that a homework assignment is due, the majority of the class period will be devoted to students presenting a subset (maybe all) of the problems that are due that day. At the end of each class period, students should submit their Daily Homework for all of the works that are due that day. Students are allowed (in fact, encouraged!) to modify their written proofs in light of presentations made in class; however, you are required to use the felt-tip pens provided in class.

  • Weekly Homework

    In addition to the Daily Homework, you will also be required to submit two formally written problems each week. You may choose any two problems marked with * that were turned in during a given week to submit the following week. The Weekly Homework assignments are subject to the following rubric:

    4This is correct and well-written mathematics!
    3This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing.
    2There is some good intuition here, but there is at least one serious flaw and/or there are too many grammatical mistakes.
    1I don’t understand this, but I see that you have worked on it; come see me!
    0I believe that you have not worked on this problem enough or you didn’t submit any work or the work is not original and came from the internet or some other external source.

    Any Weekly Homework problems that receive a score of 1, 2, or 3 can be resubmitted within one week after the corresponding problem was returned to the class. The final grade on the problem will be the average of the original grade and the grade on the resubmission. Please label the assignment as “Resubmission” on top of any problem that you are resubmitting and keep separate from any other problems that you are turning in.

    Unlike a traditional Moore method course, 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 3 homework assignments (daily or weekly) late with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. Your overall homework grade is worth 15% of your final grade.

    All of your Weekly Homework must be typed using LaTeX. LaTeX is the industrial standard for typing scientific works in mathematics, physics, computer sciences, among others. The best way to learn how to use LaTeX is just like how you learn everything else: by using it! Fortunately, there is a website called Overleaf so you can use LaTeX online for free without having to install any software. I have also created a template for your Weekly Homework that should make things much easier for you. I will try to schedule a training session during week 1 to prepare you with some basics.


  • 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 very seriously since they spearhead the work of the class.

  • The problems chosen for presentations will come from the Daily Homework. After a student has presented a problem that the class agrees is sufficient, I will often call upon another student in the audience to recap what has happened in the proof and to emphasize the salient points.

  • In order to make presentations go smoothly, presenters need to write out the proof in detail and go over the major ideas and transitions, so that he or she can make the proof clear to others.

  • The purpose of presentations is not to prove to me that the presenter has done the problem. It is to make the ideas of the proof or the solution clear to the other students.

  • Presenters need to write in complete sentences, using proper English and mathematical grammar. Here are some suggestions on how to write a proper proof.

  • Fellow students are allowed to ask questions at any point and it is the responsibility of the presenter 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.

  • Confusions and mistakes are very common when learning new mathematics and they should be handled positively to stimulate your thinking. In light of this idea, in each presentation, the presenter must do one or both of the following:
    • ask at least one question that he or she does not know the answer and then the whole class will discuss about it or
    • talk about at least one mistake that he or she made while working on the problem, how he or she understand that mistake, and then how to correct the mistake.
  • Presentations will be graded using the rubric below.

    2Completely correct and clear proof or solution.
    1Proof has technical flaws, some unclear language, or is lacking some details.
    0You were completely unprepared.

    However, you should not let the rubrics deter you from presenting if you have an idea about a proof that you would like to present, but you are worried that your proof is incomplete or you are not confident your proof is correct. You will be rewarded for being courageous and sharing your creative ideas! Yet, you should not come to the board to present unless you have spent time thinking about the problem and have something meaningful to contribute.

  • In each class, a sorted class list produced by a computer program will be shown before presentations. Students whose ranks are high in the list have higher priority to choose problems. The sorted list is not produced randomly. It takes three factors into consideration:
    • The number of past presentations: the more one presented in the past, the lower one is on the list.
    • The quality of past presentations: the better one presented in the past, the lower one is on the list.
    • Recentness of past presentations: the more recently one has presented, the lower one is on the list.
  • A student can choose not to present on a day when he or she has a high rank in the sorted list. This is called a “pass”. No one shall “pass” for two consecutive classes. If you need help to prepare presentations, see me during office hours as soon as possible.

  • In order to receive a passing grade on the presentation portion of your grade, you must present at least four times prior to each exam (2 midterms and 1 final) for a total of at least twelve times during the semester. Your grade on your presentations, as well as your level of interaction during student presentations, are worth 20% of your overall grade.


  • The object is to maintain a current account of the work we do. Every task that we encounter in the class is to be included in your portfolio. Each entry in the portfolio is intended to be complete and polished. Do not include scratch work.

  • Each of us will develop her or his own mathematical voice in this class. Not every solution will look the same. However, the form of the portfolio should be fairly standardized. It will include a cover sheet with your name on it. Begin each write-up with the statement of a task followed by your solution or proof. Some write-ups will be two lines long, others may be several pages. You can use the filler paper (the same you use for Daily Homework) to write up your solution. If you have done a perfect job in your Daily Homework, then you can just insert the page into your portfolio. Though in most cases, you will have to improve your original work.

  • The portfolio will be collected three times: Wednesday, October 2, 2019, Wednesday, November 6, 2019 and Friday, December 20, 2019, at the final exam.

  • Because you will have already know whether your solutions or proofs are correct or not by discussing them during presentations and by having them graded as Daily and Weekly Homework, portfolios will be graded solely on completeness and clarity. Clear and complete portfolios will earn a check mark, all others will be asked to resubmit within a week. Keep your portfolio current as you work, it will be too much effort to get it all organized and collated the night before it is due.

  • At the end of the semester, portfolios with three check marks will earn the full 15% possible. Two check marks will earn 10%, one check mark will earn 5% and no check marks will earn 0%.

  • In the end, you will walk away with an organized and complete collection of your work on which you can look back with pride.


There will be two midterm exams and one cumulative final exam. Each exam is worth 15% of your overall grade and may consist of an in-class portion and a take-home portion. The in-class portions of the two mid-term exams are tentatively scheduled for Wednesday, October 2, 2019 and Wednesday, November 6, 2019, and the in-class portion of the final exam is Friday, December 20, 2019. 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.


Every week, you need to write a 300 word narrative summarizing what you have learned in the past week. Use your own words to describe the main ideas. Use as few mathematical symbols as possible. Your week N journal should be accessible for students like you who have successfully finished all the work in the previous N-1 week and about to start week N. Your weekly journal shall be submitted electronically at Canvas. Once in a while, you will also be given some topics that you need to discuss in your weekly journal.


Your final grade will be determined by the scores of your homework, presentations/participation, portfolio, journals, and exams.

Midterm Exam 115%
Midterm Exam 215%
Final Exam15%
Letter GradePercentage Score
A92% ≤ score
A-90% ≤ score < 92%
B+88% ≤ score < 90%
B82% ≤ score < 88%
B-80% ≤ score < 82%
C+78% ≤ score < 80%
C72% ≤ score < 78%
C-70% ≤ score < 72%
D+68% ≤ score < 70%
D60% ≤ score < 68%
Fscore < 60%

Academic Integrity

I have zero tolerance on dishonesty. Any forms of dishonesty such as copying homework or cheating on quizzes and examinations, will result in zero credit for that particular assignment, and will be reported to the Academic Standards Committee. The highest penalty a student can receive is “F for cheating” for the course. There might be additional sanctions by the Academic Standards Committee such as dismissal from the college. See Utica College official page for Academic Honesty for more details.

Special Accomodation

Any student who has need of special accommodations in this class due to a documented disability should speak with me as soon as possible, preferably within the first two weeks of class. You should also contact Judy Borner, Director of Learning Services in the Academic Support Services Center (315-792-3032 or ) in order to determine eligibility for services and to receive an accommodation letter. We will work with you to help you in your efforts to master the course content in an effective and appropriate way. See Utica College official page for Office of Learning Services.


It is the students’ responsibility to keep informed of all announcements, syllabus adjustments, or policy changes during the semester via this web page or via school emails. The author of this syllabus reserves the right to change it with notice at any time during the semester.