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

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

Office hours 
Tu, Th 12:001:00 in Math 4101B 
Tu, Th 11:301:00 in Math 5107 
Recitation 
R01: MW 11:0011:53 Library E4320 R02: TuTh 2:30 3:23 Physics P129 
MW
11:0011:53 P131 Starts after 10/2 
TA's 

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
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 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 Javits 103.
First recitation on Tues., 8/27.
During joint lectures through 10/2, students with last names starting
AO attend recitation in Library
E4320, students with last names PZ 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.
1. 
8/27 Joint class: Introduction, Sets, ordered pairs and functions (Ebin) 
Read pages 122 
2. 
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 
3. 
9/3 Joint class: The real numbers are a complete ordered field. The Archimedean property. (Ebin) 
Read
pages 2336; 
4. 
9/10 Joint class: Infinity, unboundedness. Intro to sequences. (Ebin) 
Read pages 2838 
5. 
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, 
6. 
9/17 Joint class: Limit laws for sequences. (Li) 
Read pages 3955 
7. 
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 
8. 
9/24 Joint class: Monotone and Cauchy sequences. (Li) 
Read pages 5676 
9. 

No HW

Joint
Midterm I 9/26
in the lecture room, Javits 103 





10. 
Everything from here on is for MAT320 only 

11. 
Limsup and Liminf, BolzanoWeierstrass, Metric spaces and R^{n} as a metric space 
Read pages 7784

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

13. 
Read pages 8598 

14. 
HW
due 10/23:
3.7, problems 5, 6b, 7, 11, 13; 4.1, problems 4, 6, 12bd, 13,
17 

15. 

Read pages 90115 
16. 

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

17. 
Read pages 116129 

18. 
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 130145, skip pages 145 bottom half through 152 

19. 

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 nonuniform 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. 
20. 


21. 

No HW due November 13. 
22. 

HW
due 11/20 5.6, problems 3, 9, 10;
6.1, problems 2, 7, 9; 6.2, problems 5, 12, 15, 19 
23. 

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 198208 
24. 

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 nonnegative,
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 
25. 
Final Exam: Thurs. Dec.
19, 8:00am10:45am
Practice
final for 319
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