Fall 2019 MAT 319: Foundations of Analysis

Fall 2019 MAT 320: Introduction to Analysis


TuTh 10:00-11:20 Library E4310 (CHANGE) (through 10/2: joint lectures in Math P-131)

TuTh 10:00-11:20 Javits Lecture 103, then Math P-131


Yu Li

David Ebin

Office hours

Tu 11:30-12:30 in Math P-143, Tu 2:00-3:30 and Th 11:30-12:00 in Math 3-109

Tu, Th 11:30-1:00 in Math 5-107


MW 11:00-11:53 Harriman 112

MW 11:00-11:53 P-131


Apratim Chakraborty

Owen Mireles Briones

Office hours

MW 2:00-3:00, Th 4:00-5:00 in MLC

W 4:00-6:00 in MLC, W 3:00-4:00 in Math 2-105


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 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 collected in Wednesday recitation. 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%.

Syllabus/schedule (subject to change)
All joint lectures through 10/2 meet in Math P-131.
First recitation on Wed 8/27, second recitation Wed 9/3.
During joint lectures through 10/2, students with last names starting A-O attend recitation in Harriman 112, students with last names P-Z attend recitation in Lgt Engr Lab 152

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, motivation: what are real numbers? (Ebin)

Read pages 1-19


Joint class: Properties of numbers; induction; concept of a field. (Ebin)

HW due 9/3: 1.3, 1.4, 1.10, 1.12, 2.2, 2.5, 3.1, 3.4, 3.6

No class: day after Labor Day


Joint class: Completeness axiom for real numbers; Archimedean property. (Ebin)

Read pages 20-27;
HW due 9/10: parts eghimr of: 4.1,4.2,4.3,4.4; and 4.8,4.10,4.11,4.12,4.14


Joint class: Infinity, unboundedness. Intro to sequences. (Ebin)

Read pages 28-38


Joint class: Limit of a sequence. (Ebin)

HW due 9/17: 5.2, 5.6, 7.3, 7.4, 8.1ac


Joint class: Limit laws for sequences. (Grushevsky)

Read pages 39-55


Joint class: Divergence to infinity, more formal proofs. (Grushevsky)

HW due 9/24: 8.3, 8.6, 8.8, 8.10, 9.1, 9.3, 9.5, 9.12, 9.14


Joint class: Monotone and Cauchy sequences. (Grushevsky)

Read pages 56-65


Joint class: Subsequences. (Grushevsky)

No HW: prepare for the midterm

Joint Midterm I in Math P-131.

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


Joint class: Subsequences. (Grushevsky)

HW due 10/8: 10.1, 10.2, 10.5, 10.8, 10.9, 11.2, 11.4, 11.5, 11.8, 11.9

Everything from here on is for MAT320 only


Limsup and Liminf, Bolzano-Weierstrass, Metric spaces and Rn as a metric space

  Read pages 66-77


HW due 10/15:12.1, 12.2, 12.4, 12.5, 12.9ab,  12.10,  12.14, 13.1, 13.3, 13.4


Read pages 78-87


HW due 10/22: nothing due this week


Read pages 90-104


HW due 10/29: 13.8b, 13.9, 13.11, 13.12, 13.14, 14.1ace, 14.3ace, 14.6, 14.12, 14.13


Read pages 105-122


HW due 11/5:  15.2, 15.3, 15.7, 16.4acd, 16.9, 17.1ac, 17.2, 17.4, 17.8, 17.14


Second midterm on November 13
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: Friday December 12, 11.15AM-1.45PM
Practice final for 319

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