Spring 2021 MAT 319: Foundations of Analysis

Spring 2021 MAT 320: Introduction to Analysis

Lecture Schedule

TuTh 11:30-12:50 (through 2/25: joint lectures online)

TuTh 11:30-12:50 online


Dimitrios Ntalampekos

David Ebin

Office hours

Tu, Thu 1:00-2:00; MLC hour Tu 3:00-4:00

Tu, Th 10:00-11:30


MW 2:40-3:35; R01  in Engineering 145, R02 and R03 online

MW 2:40-3:35 online


Mohamad Rabah and Yevgeniya Zhukova

Jacob Mazor

Office hours

Rabah: Fri 10:30-11:30; MLC hours Th 5:30-6:30, Th 6:30-7:30

Zhukova: TBA



A careful study of the theory underlying topics in one-variable calculus, with an emphasis on those topics arising in high school calculus. The real number system. Limits of functions and sequences. Differentiations, integration, and the fundamental theorem. Infinite series.

A careful study of the theory underlying calculus. The real number system. Basic properties of functions of one real variable. Differentiation, integration, and the inverse theorem. Infinite sequences of functions and uniform convergence. Infinite series.


The purpose of this course is to build rigorous mathematical theory for the fundamental calculus concepts, sequences and limits, continuous functions, and derivatives. We will rely on our intuition from calculus, but (unlike calculus) the emphasis will be not on calculations but on detailed understanding of concepts and on proofs of mathematical statements.

An introductory course in analysis, required for math majors. It provides a closer and more rigorous look at material which most students encountered on an informal level during their first two semesters of Calculus. Students learn how to write analysis proofs. Students (especially those thinking of going to graduate school) should take this as early as possible.


C or higher in MAT 200 or permission of instructor; C or higher in one of the following: MAT 203, 205, 211, 307, AMS 261, or A- or higher in MAT 127, 132, 142, or AMS 161.
Math majors are required to take either MAT 319 or MAT 320


Robert G. Bartle and Donald R. Sherbert,  Introduction to Real Analysis, 4th edition


Weekly problem sets will be assigned, and should be uploaded to Blackboard. The emphasis of the course is on writing proofs, so please try to write legibly and explain your reasoning clearly and fully. You are encouraged to discuss the homework problems with others, but your write-up must be your own work.
Late homework will never be accepted, but under documented extenuating circumstances the grade may be dropped.


Homework: 20%, Midterm I: 20%, Midterm II: 20%, Final: 40%.

Exam Policy

The two Midterms and the Final exam will be administered synchronously through Blackboard. You will have to write clearly, take a clear photo or scan of your work, and upload it to Blackboard. Note that credit cannot be given for manuscripts that are not legible and it is your responsibility to upload a readable copy of your work in time. In order to eliminatethe impact of internet disruptions, sufficient extra time will be given for uploading your work to Blackboard. The exams will be proctored through Zoom by the instructor and the teaching assistants. Every student who is taking an exam will join the Zoom session of the class with camera enabled. Minor internet disruptions during proctoring are permissible

Syllabus/schedule (subject to change)
All joint lectures through 2/25 are online.
First recitation on Wed 2/3, second recitation Wed 2/10.
Recommendations on choosing MAT 319 vs MAT 320 will be made based upon your performance on the first midterm and homework to that date.


Joint class: Introduction, sets and functions (Ntalampekos)

Read Chapter 1


Joint class: Well ordering property and mathematical induction; finite countable and uncountable sets, Cantor's theorem (Ntalampekos)

HW due 2/10: page 10, problems 5, 10, 16, 23, 24,  


Joint class: Field properties and order properties of real numbers; the completeness property. (Ntalampekos)

Read pages 23-39
HW due 2/17: page 15, problems 3, 5, 7, 11; page 22, problems 4, 6, 12


Joint class: Applications of the completeness property and intervals (Ntalampekos)

Read pages 40-53


Joint class: Limit of a sequence and theorems about limits (Ebin)

Read pages 54-69

HW due 2/24: page 44, problems 3, 5, 7, 9, 13, 17; page 52, problems 3, 7, 9, 14


Joint class: Monotone sequences. (Ebin)

Read pages 70-77


Joint class: Subsequences and the Bolzano-Weierstrass theorem (Ebin)



Joint class: Cauchy sequences. (Ebin)

Read pages 78-93


No HW: prepare for the midterm

Joint Midterm I, March 2, 11:30-12:50 via Blackboard

Practice midterm 1, Practice midterm 2, Practice midterm 2 solutions


Everything from here on is for MAT320 only

HW due 3/10: page 91, problems 1, 2, 4, 5, 10, 14


Properly divergent sequences and infinite series

  Read sections 3.6 and 3.7










Second midterm
Possible topics for the exam:
equivalence relations and equivalence classes; natural numbers, integers, rational numbers, algebraic numbers, real numbers (a complete ordered field) and complex numbers; absolute value;max, min sup and inf for subsets of the real numbers; Archimedian property; positive numbers have square roots; sequences and series and their properties; Bolzano Weierstrass theorem; inner product and norm for R^n; Schwartz inequality; metric spaces; R^n as a metric space; completeness for metric spaces;  Compactness; Heine-Borel theorem;  Is a bounded complete metric space necessarily compact; open and closed sets in a metric space; ratio test for convergence of series; harmonic series; convergence of alternating series; exponential function of a complex variable called E(z); E(z+w) = E(z) E(w); sine and  cosine from E(ix); Continuity of a function from one metric space to another;


Read pages 126 -143


No HW due November 12.  review for exam


HW due 11/19  17.15,  18.3, 18.5a, 18.9, 18.12b


Read pages 145-154 We did not do all of this in classs because it is rather routine, but you are responsible for it


HW  due 11/24 19.1acde, 19.4, 19.7, 20.14, 20.17


  Read pages 205-220 and 243-265

Final Exam: Tuesday May 18, 11.15AM-1.45PM
Practice final for 319

Disability Support Services: If you have a physical, psychological, medical, or learning disability that may affect your course work, please contact Disability Support Services (DSS) office: ECC (Educational Communications Center) Building, room 128, telephone (631) 632-6748/TDD. DSS will determine with you what accommodations are necessary and appropriate. Arrangements should be made early in the semester (before the first exam) so that your needs can be accommodated. All information and documentation of disability is confidential. Students requiring emergency evacuation are encouraged to discuss their needs with their professors and DSS. For procedures and information, go to the following web site http://www.ehs.sunysb.edu and search Fire safety and Evacuation and Disabilities.

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Critical Incident Management: Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, and/or inhibits students' ability to learn.