Spring
2021 MAT 319: Foundations of Analysis 
Spring
2021 MAT 320: Introduction to Analysis 

Lecture
Schedule 
TuTh
11:3012:50 (through 2/25: joint lectures online) 
TuTh
11:3012:50 online 
Instructor 

Office
hours 
Tu,
Thu 1:002:00; MLC hour Tu 3:004:00 
Tu,
Th 10:0011:30 
Recitation 
MW
2:403:35; R01 in Engineering 145, R02 and R03 online 
MW
2:403:35 online 
TA 

Office
hours 
Rabah: Fri 10:3011:30; MLC hours Th 5:306:30, Th 6:307:30 Zhukova: TBA 
TBA

Description 
A
careful study of the theory underlying topics in
onevariable 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.

Overview 
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.

Prerequisites 
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.


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

Homework 
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 writeup must be
your own work. 

Grading 
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.
1. 
Joint
class: Introduction, sets and functions (Ntalampekos) 
Read
Chapter 1 
2. 
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, 
3. 
Joint
class: Field properties and order properties of real numbers;
the completeness property. (Ntalampekos) 
Read
pages 2339 
4. 
Joint
class: Applications of the completeness property and intervals
(Ntalampekos) 
Read
pages 4053 
5. 
Joint
class: Limit of a sequence and theorems about limits (Ebin) 
Read pages 5469 HW
due 2/24: page 44, problems 3, 5, 7, 9, 13, 17; page
52, problems 3, 7, 9, 14 
6. 
Joint
class: Monotone sequences. (Ebin) 
Read
pages 7077 
7. 
Joint
class: Subsequences and the BolzanoWeierstrass theorem (Ebin) 

8. 
Joint
class: Cauchy sequences. (Ebin) 
Read
pages 7893 
9. 

No
HW: prepare for the midterm 
Joint
Midterm I,
March 2, 11:3012:50 via Blackboard 
Practice midterm 1,
Practice midterm 2,
Practice
midterm 2 solutions 

10. 
Everything from here on is for MAT320 only 
HW due 3/10: page 91, problems 1, 2, 4, 5, 10, 14 



11. 
Properly
divergent sequences and infinite series

Read sections 3.6 and 3.7 
12. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 

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; HeineBorel 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; 
20. 
Read
pages
126
143 

21. 

No HW due November 12. review for exam 
22. 

HW due 11/19 17.15, 18.3, 18.5a, 18.9, 18.12b 
23. 

Read
pages 145154 We did not do all of this in classs because it
is rather routine, but you are responsible for it 
24. 

HW due 11/24 19.1acde, 19.4, 19.7, 20.14, 20.17 
25. 
Read
pages 205220 and 243265 
Practice final for 319
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