Fall
2019 MAT 319: Foundations of Analysis 
Fall
2019 MAT 320: Introduction to Analysis 

Schedule 
TuTh
10:0011:20 Library E4310 (CHANGE) (through 10/2:
joint lectures in Math P131) 
TuTh
10:0011:20 Javits Lecture 103, then Math P131 
Instructor 

Office
hours 
Tu
11:3012:30 in Math P143, Tu
2:003:30 and Th 11:3012:00 in
Math 3109 
Tu,
Th 11:301:00 in Math 5107 
Recitation 
MW
11:0011:53 Harriman 112 
MW
11:0011:53 P131 
TA 

Office
hours 
MW
2:003:00, Th 4:005:00 in MLC 
W
4:006:00 in MLC, W 3:004:00 in Math 2105 
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 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 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 writeup must be
your own work. 

Grading 
Homework:
20%, Midterm I: 20%, Midterm II: 20%, Final: 40%. 
Syllabus/schedule (subject to change)
All joint lectures through 10/2
meet in Math
P131.
First recitation on Wed 8/27, second recitation
Wed 9/3.
During joint lectures through 10/2, students with last names starting
AO
attend recitation in Harriman 112, students with last names PZ 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.
1. 
Joint
class: Introduction, motivation: what are real numbers? (Ebin) 
Read
pages 119 
2. 
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 

3. 
Joint
class: Completeness axiom for real numbers; Archimedean
property. (Ebin) 
Read
pages 2027; 
4. 
Joint
class: Infinity, unboundedness. Intro to sequences. (Ebin) 
Read
pages 2838 
5. 
Joint
class: Limit of a sequence. (Ebin) 
HW
due 9/17:
5.2, 5.6, 7.3, 7.4, 8.1ac 
6. 
Joint
class: Limit laws for sequences. (Grushevsky) 
Read
pages 3955 
7. 
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 
8. 
Joint
class: Monotone and Cauchy sequences. (Grushevsky) 
Read
pages 5665 
9. 
Joint
class: Subsequences. (Grushevsky) 
No
HW: prepare for the midterm 
Joint
Midterm I
in Math P131. 
Practice
midterm 1, Practice
midterm 2, Practice
midterm 2 solutions 

10. 
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 

11. 
Limsup and Liminf, BolzanoWeierstrass, Metric spaces and R^{n} as a metric space 
Read pages 6677 
12. 
HW
due 10/15:12.1,
12.2, 12.4, 12.5, 12.9ab, 12.10, 12.14, 13.1,
13.3, 13.4 

13. 
Read pages 7887 

14. 
HW
due 10/22:
nothing due this week 

15. 

Read pages 90104 
16. 

HW
due 10/29:
13.8b, 13.9, 13.11, 13.12, 13.14, 14.1ace, 14.3ace, 14.6,
14.12, 14.13 
17. 
Read pages 105122 

18. 
HW
due 11/5: 15.2,
15.3, 15.7, 16.4acd, 16.9, 17.1ac, 17.2, 17.4, 17.8, 17.14 

19. 

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; 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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