Fall 2014 MAT 319: Foundations
of Analysis 
Fall 2014 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 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 P143 
Recitation 
MW 11:0011:53 Harriman 112 
MW 11:0011:53 Lgt
Engr Lab 152 
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 
Kenneth Ross Elementary
Analysis: The Theory of Calculus, 2nd 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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