Fall 2019 MAT 319: Foundations of Analysis

Fall 2019 MAT 320: Introduction to Analysis


TuTh 10:00-11:20 

Javits 103 (through 10/2: joint lectures in Javits 103)

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

(through 10/2: joint lectures in Javits 103)


Yu Li

David Ebin

Office hours

Tu, Th 12:00-1:00 in Math 4-101B

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


R01: MW 11:00-11:53 Library E4320

R02: TuTh 2:30- 3:23 Physics P129

MW 11:00-11:53 P-131 Starts after 10/2


James Seiner

Ruijie Yang

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 or Thursday 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 Javits 103.
First recitation on Tues., 8/27.
During joint lectures through 10/2, students with last names starting A-O attend recitation in
Library E4320,  students with last names P-Z attend recitation in Physics P129
Recommendations on choosing MAT 319 vs MAT 320 will be made based upon your performance on the first midterm and homework to that date.


8/27 Joint class: Introduction, Sets, ordered pairs and functions (Ebin)

Read pages 1-22


8/29 Joint class: Induction. Finite, countable and uncountable sets, Cantor's theorem (Ebin)

HW due 9/4: 1.1, problems 5, 10 and 16; 1.2 problems 2, 6 and 18;  1.3, problems 4, 10 and 12


9/3 Joint class: The real numbers are a complete ordered field.  The Archimedean property. (Ebin)

Read pages 23-36;
HW due 9/11: 2.1, problems 2bc, 4, 8, 20; 2.2, problems 4, 7, 16; 2.3, problems 6, 12


9/10 Joint class: Infinity, unboundedness. Intro to sequences. (Ebin)

Read pages 28-38


9/12 Joint class: Limit of a sequence. (Ebin)

HW due 9/18: 2.4, problems 3, 13, 15, 19; 2.5, problems1, 3, 6, 8, 11; 3.1, problem 3ac,


9/17 Joint class: Limit laws for sequences. (Li)

Read pages 39-55


9/19 Joint class: Divergence to infinity, more formal proofs. (Li)

HW due 9/25: 3.1, problems 4, 8, 12; 3.2, problems 4, 10, 11, 13, 17, 21, 20


9/24 Joint class: Monotone and Cauchy sequences. (Li)

Read pages 56-76



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


Joint Midterm I

9/26 in the lecture room, Javits 103


Everything from here on is for MAT320 only


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

  Read pages 77-84

HW due 10/9 3.3, problems 6, 8, 9, 11; 3.4, problems 4, 5, 9, 10, 16, 19



HW due 10/16: 3.5, problems 2, 4, 7, 9, 11, 12, 14; 3.6, problems 5, 6, 10



Read pages 85-98



HW due 10/23: 3.7, problems 5, 6b, 7, 11, 13; 4.1, problems 4, 6, 12bd, 13, 17


Read pages 90-115


HW due 10/30: 4.2, problems 5, 9, 12, 13, 14, 15; 4.3, problems 1, 7, 11; 5.1, problem 3   



Read pages 116-129



HW due 11/6:  5.1, problems 6, 12; 5.2, problems 3, 8, 10, 14; 5.3, problems 6, 18; 5.4, problems4, 8

Read pages 130-145, skip pages 145 bottom half through 152


Second midterm on November 12
Possible topics for the exam: Sets and functions, Finite and infinite sets.  Infinite sets may be countable or uncountable.  Prove that the rationals are countable and the reals are not.  The real numbers are a complete ordered field.  Understand the words complete, ordered and field. Absolute value and the triangle inequality. Intervals and the nested intervals theorem.   Sequences  and their limits.  Monotone and strictly monotone sequences, bounded sequences, subsequences and the Bolzano Weierstrass theorem.  Cauchy sequences.  Use the completeness property of the reals to prove that every Cauchy sequence converges.  Infinite series, geometric series.  Prove that the harmonic series does not converge, Alternating series and Cauchy criterion for series, Comparison tests for series.  Continuous functions.  Uniform and non-uniform continuity.  Prove that a continuous function on a closed bounded interval is uniformly continuous.  Monotone and inverse functions. Jumps for monotone functions.  Continuous inverse theorem. Definition and basic properties of derivatives. Using Caratheodory's theorem, prove the chain rule for differentiable functions and use the chain rule to find the derivative of an inverse function.   




No HW due November 13.


HW due 11/20  5.6, problems 3, 9, 10; 6.1, problems 2, 7, 9; 6.2, problems 5, 12, 15, 19


HW due 12/2 6.3, problems 2, 5, 6; 6.4, problems 2, 8, 12, 16; 7.1, problems 4, 7, 11 Read pages 198-208


Possible topics for the final on December 19: everything above for the practice midterm I, and for the second midterm, the mean value theorem and its uses, L'Hospital's rule, Taylor's theorem with remainder and its use to find relative extrema, definition of a convex function, a differential function on an interval is convex iff its second derivative is non-negative, partitions of an interval and tagged partitions, Riemann sums and Riemann integrals and basic properties of the integral, proof that an integrable function must be bounded, step functions, proof that every step function is integrable, the squeeze theorem and the proof that every integrable function can be "squeezed" by step functions, proofs that every continuous and every monotonic function is integrable, both forms of the fundamental theorem of calculus,  pointwise and uniform convergence of sequences of functions, the uniform norm for functions, proof that the uniform limit of continuous functions is continuous, but that the pointwise limit may not be, interchange of the integral of the uniform limit of a sequence of functions with the limit of the integrals, proof that on any bounded interval the exponential function is a uniform limit of polynomials and construction of the polynomials



Final Exam: Thurs. Dec. 19, 8:00am-10:45am
Practice final for 319

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