MAT 320: Introduction to Analysis, Fall 2021

Dept of Mathematics , Stony Brook University

MAT 319 and 320

MAT 319 (Foundations of Analysis) and MAT 320 are taught as a single lecture until the first midterm. Based on the results of the first midterm, students may switch into MAT 320, which will proceed at a faster pace than MAT 320. Professor Martens will teach MAT 319, and the two of us will share lectures up until the first midterm.

MAT 320 Instructor

        Prof. Christopher Bishop,
        Dept. Phone: (631)-632-8290
        Dept. FAX: (631)-632-7631
        my homepage

MAT 319 Instructor

        Prof. Marco Martens, his homepage

Course Summary

This is a course in real analysis or roughly "calculus with proofs". We will start by recalling set notation, the principle of induction, and the basic properties of finite and infinite sets. We then discuss the axioms of the real numbers, and learn to rigorously derive various familiar properties. When then study sequences and limits. Around this point MAT 319 and 320 will separate. In MAT 320 will will then discuss continuity, differentiation, Riemann integration, sequences of functions and infinite series. If time permits, we will discuss basic point set topology, metric spaces, the generalized Riemann integral and the Lebesgue integral.

Time and Place

For the first few weeks, MAT 319 and MAT 320 both meet Tuesdays and Thursdays 9:45am-11:05am in Stony Brook Union room 103-02. After the classes split MAT will continue in that room and MAT 320 will meet in Mathematics P-131 (this room is directly behind the elevators on the PL level of the Math Tower). Recitations meet twice a week as follows:
        MAT 319 R01 MW 11:45am-12:40pm Earth-Space Science 181, TA is Jordan Rainone
        MAT 319 R02 MW 11:45am-12:40pm Frey Hall 309, TA is Paul Sweeney
        MAT 319 R03 MW 11:45am-12:40pm Socbehav Sci N310, Dylan Galt
        MAT 320 R01 MW 11:45am-12:40pm Physics P116, TA is Jordan Rainone (this meets after 1st midterm)
Note that all recitations meet at the same time.

Office Hours

TuTh 11:10-12:40 (after class) and by appointment. Meetings on other days can be in person or via Zoom, depending on my schedule.


The textbook is "Introduction to Real Analysis, 4th Edition" by Robert Bartle and Donald Sherbert.


Grades will be based on weekly problem sets, two midterms and a final exam. Each component will count for 25% of the total grade.


Final Exam

The final exam is Tuesday, December 14, 8:00am-10:45am. Location will be be announced later. Possibly all final exams will be online.


        Blackboard is the Stony Brook University class management system. Your homework, quiz and exam grades will be posted here. Letter grades for the course are posted in the Solar System. I will occasionally post announcements in Blackboard; you should receive email notifications whenever this occurs.

Solar System

        Solar System is the Stony Brook University administrative management system (registration, bills,...). It is not used for classes, except to post letter grades at the end of the semester..

Stony Brook Gmail

        Check your email here.

Stony Brook Virtual SINC Site

        The Virtual Sinc Site gives you access to various software packages on a university license, such as Mathematica and Matlab. I don't plan to use any of these in this class, but this resource may be helpful in other classes. Using the virtual Sinc Site requires downloading the Citrix receiver software (you will be prompted).

Important University Dates

    Link to university academic calendars, including final exam calendars.

    First day of classes: Monday August 23, 2021.
    Labor Day, no classes: Monday Sept 6, 2021.
    Fall break, No classes Mon Oct 11 and Tue Oct 12
    Thanksgiving break: Wed November 24 to Sunday November 28, 2021.
    Last day of classes: Monday December 6, 2021.
    Reading day: Tuesday December 7, 2021.
    Finals: Wednesday December 8 to Thursday December 16, 2021.
    MAT 320 Final Exam: 8:00am-10:45am Tuesday, Dec 14, 2021
    Commencement: Friday December 17, 2021

University deadlines

See the following page for deadlines for things like withdrawing from classes without penalty, applying for P/NC, changing major, ... University deadlines

Pass/No Credit

This policy allows you to set a threshold so that if you score above the threshold in a class you get a that grade on your transcript, and otherwise you get a P (for pass) or NC (no credit), neither of which affects your GPA. For example, if you set the threshold at C and if you get a C- or a D you will get a P in the course (which means it won't count towards major requirements, but also won't affect your GPA). A grade of F gives an NC (also won't affect your GPA), and any grade equal to or higher than your threshold will count as usual. Generally, only one course per semester may be designated P/NC. Check the Bulletin for precise dates and requirements; your major may also have rules about which classes these may be used for. SBU G/P/NC page.

Problem sets

There will be a problem set due each week (except the first week of class and midterm weeks; sections covered the week before a midterm are due the week after the midterm). These should be handed in at the first recitation of the week. Problems will be taken from the textbook and are listed in the lecture schedule below. Problems from a section are due in recitation the week after that section is covered in lectura (except for midterm weeks).

Most of the problems are short proofs. In general, a few sentences will be sufficient for most problems. A proof should be a clearly written argument; the graders may penalize organizational or grammatical errors that make the problem hard to read or the reasoning hard to follow. Include a diagram or picture if this helps convey the meaning. For most problems, I would expect that a couple of practice drafts will be necessary. It is hard even for an experienced mathematician to write down a complete, legible proof off the top of their heads (published proofs can take weeks, months or even years of polishing to make the ideas understandable, but I would guess 15-30 minutes would be sufficient for most problems in this course).

Tentative Lecture Schedule and Assigned Problems

Week 1, Aug 23 - Aug 27
        Topics covered: Aug 24 notes , Aug 26 notes
        1.1 Sets and functions (problems 6, 14, 18, 21)
        1.2 Induction (problems 7, 11, 17)
        1.3 Finite and infinite sets (problems 4, 12, 13)
        A few people did not have the textbook the first week. Here is a scan of problems due Aug 30 from textbook sections 1.1, 1.2, 1.3 ,

Week 2, Aug 30 - Sept 3. Class canceled Thursday 9/2 due to hurricane damage.
        Since there is no recitation Monday 9/6, Problem set 2 is due in recitation on Wed., 9/8. Section 2.1 problems only.
        Topics covered:
        2.1 Algebraic and order axioms (problems 4, 8, 17, 19)

Week 3, Sept 6 - Sept 10
        Topics covered: Sept 7 notes , Sept 9 notes
        Revised problem assignment for Week 3 is given below.
        2.2 Absolute value and the real line (problems 17)
        2.3 The completeness property (problems 10, 12,13)
        2.4 Applications of the Supremum property (problems 5, 7, 11, 19)
        2.5 Intervals (no problems assigned)

Week 4, Sept 13 - Sept 17
        Starting this week you need only turn in the problems marked with a *.
        The others are recommended to try.
        Some will be chosen for the midterm, so you should be able to do them.
        Problems are still due in recitation on Mondays.

        Topics covered:
        3.1 Sequences and their limits (problems 4, 5*, 11*, 17)
        3.2 Limit theorems (problems 2*, 7*, 15, 20)
        3.3 Monotone sequences (problems 2, 7*, 9)

Week 5, Sept 20- Sept 24:
        Topics covered: Sept 21 notes , Sept 23 notes
        3.4 Bolzano-Weierstrass theorem (problems 6, 9, 18*)
        3.5 The Cauchy Criterion (problems 4*, 6, 11*)
        3.6 Properly divergence sequences (problems 2, 5, 10)
        3.7 Introduction to infinite series (problems 5, 11*, 15*, 17)

Week 6, Sept 27 - Oct 1
        Topics covered:
        Tuesday Sept 28, Review of Chapters 1-3
        Thursday Sept 30, Midterm 1 in lecture

Week 7, Oct 4 - Oct 8
        MAT 319 and 320 will meet together on Tuesday to discuss the midterm and splitting class.
        MAT 320 moves to Math Tower P-131 staring Thursday, October 7.

        Starred homework from Sections 4.1 to 5.1 due in recitation Wednesday Oct 13.
        4.1 Limits of functions Topics covered: Oct 5 notes , Oct 7 notes
        4.1 Limits of functions (problems 6, 13*, 16)
        4.2 Limit Theorems (problems 8, 12*, 14)
        4.3 Extensions of the limit concept - read on your own (problems 7*, 11)
        5.1 Continuous functions (problems 3, 6, 12*, 14*, 15)

Week 8, Oct 11 - Oct 15, Fall break, no classes Oct 11-12 (no recitation Monday, no lecture Tuesday)
        Topics covered: Oct 14 notes
        5.2 Combinations of continuous functions (problems 3*, 8*, 11, 14)
        5.3 Continuous functions on intervals (problems 6, 13*, 18)

Week 9, Oct 18 - Oct 22
        Topics covered: Oct 19 notes , Oct 21 notes
        5.4 Uniform continuity (problems 2, 6*, 15, 16*)
        Short (but not simple) proof of Weierstrass approximation theorem
        Sketch of Weierstrass's proof of his approximation theorem
        Bernstein's proof of Weierstrass approximation theorem
        5.6 Monotone and inverse functions (problems 2, 9, 10, 12*, 13)
        6.1 The deriviative (problems 7, 9, 13*, 17)
        6.2 The mean value theorem (problems 8, 11, 12, 13*, 15)
        A proof that the Weierstrass function is nowhere differentiable is in Section 5.2 of my book Fractals in probability and analysis with Y. Peres.

Week 10, Oct 25 - Oct 29
        Topics covered: Oct 26 notes , Oct 28 notes .
        6.4 Taylor's theorem (problems 8*, 12, 16)
        7.1 The Riemann integral (problems 7, 8*)
        7.2 Riemann integrable functions (problems 6* (give an example), 8, 9, 15*)

Week 11, Nov 1 - Nov 5
        Topics covered: Nov 2 notes . Nov 4 notes .
        7.3 The fundamental theorem (problems 8, 14, 16*, 21*, 22*, due Mon, Nov 15)
        Appendix C, The Riemann and Lebesgue criteria
        A proof of Theorem 8.2.5 (omitted from textbook) .

Week 12, Nov 8 - Nov 12
        Topics covered:
        Nov 9 notes
        Sample Midterm 2
        Midterm 2 (covers chapters 4-7) Histogram of Midterm 2 total scores , Scatter plot of T/F versus Proof scores ,

Week 13, Nov 15 - Nov 19
        Topics covered:
        Nov 16 notes
        Nov 18 notes
        8.1 Pointwise and uniform convergence (problems 12, 19, 21, 24*)
        8.2 Interchange of limits (problems 3*, 14, 16*, 17, 18)
        9.1 Absolute convergence (problems 2*, 7, 8*, 9, 15, 16)
        9.2 Tests for absolute convergence (problems 15, 17, 19)

Week 14, Nov 22 - Nov 26
        Thanksgiving break, no class Thursday.
        Tuesday's class in on Zoom. See Blackboard for the link.
        Nov 23 notes for online lecture
        No homework over Thanksgivng break
        9.3 Tests for non-absolute convegence (problems 6, 9, 10, 12)
        9.4 Series of functions (problems 2, 7, 11, 12, 15)

Week 15, Nov 29 - Dec 3 Last week of classes. Recitation meet Mon Dec 6.
        Tuesday's class will be online. Zoom link is on class Blackboard page. I will also email to class.
        Topics covered:
        Nov 30 notes
        online slides for Tuesday
        11.1 Open and closed sets (problems 10, 11, 16}
        11.2 Compact sets (problems 4, 9, 11)
        11.3 Continuous functions (problems 2, 7, 10)
        11.4 Metric spaces (problems 7, 9, 12)

Final Exam on Tuesday Dec 14, 8:00am-10:45pm in usual lecture room, P-131 Math Tower.

        Sample final: sample 320 final
        Solutions for final: answer key

Summer internship at Scripps in La Jolla, CA

        email soliciting applications , deadline Jan 14, 2022.
        program website

Topics in the history of Calculus

Below are some reading about the history of calculus that may be of interest.
        Wikipedia page on calculus.
        Wikipedia page on Issac Newton.
        Wikipedia page on Newton-Leibniz controversy.
        Wikipedia page on the discovery of the planet Neptune (using only mathematics).
        Wikipedia page on Gauss.
        Wikipedia page on Riemann.
        Wikipedia page on Lebesgue.

Rankings of math departments - 2020

        The 2021 Shanghai Ranking of mathematics departments around the world. Stony Brook was placed 19th in the world and 10th in the United States.

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