Fall 2018 MAT 319: Foundations of AnalysisFall 2018 MAT 320: Introduction to Analysis
ScheduleTuTh 10:00-11:20 Library E4320TuTh 10:00-11:20 Math P-131 ( through 10/2: joint lectures in Library E4320)
InstructorLisa BergerSamuel Grushevsky
Office hoursTu 1.30-2.30, Th 12.30-1.30 in Math 4-100A, Tu 11.30-12.30 in Math P-143TuTh 11.30-12.30, W 1.30-2.30 in Math 3-109
RecitationMW 11.00-11.53 Library E4320MW 11.00-11.53 Math P-131
TAPrithviraj ChowdhuryJack Burkart
Office hoursW 2-3, Th 1-2 in Math 5-125A, W 3-4 in MLCMW 2.30-3.30 in Math 2-105, Tu 2.30-3.30 in MLC
Description 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.
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, 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.
Math majors are required to take either MAT 319 or MAT 320
TextbookKenneth Ross Elementary Analysis: The Theory of Calculus, 2nd edition
Homework Weekly problem sets will be assigned, and must be handed in in person 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 write-up must be your own work.
Late homework will never be accepted, but under documented extenuating circumstances the grade may be dropped. Announced and unannounced quizzes may be given during the lectures or during the recitation sections, and additional in-class work may be completed and graded. Missed quizzes and in-class work may not be made up.Your lowest homework or quiz grade will be dropped at the end of the class.
GradingHomework: 20%, Midterm I: 20%, Midterm II: 20%, Final: 40%.

Syllabus/schedule (subject to change)
All joint lectures through 10/2 meet in Library E4320.
First recitation on Wed 8/29, second recitation Wed 9/5.
During joint lectures through 10/2, students with last names starting A-L attend recitation in Library E4330, students with last names M-Z attend recitation in Math P-131

Recommendations on choosing MAT 319 versus MAT 320 will be made based upon your performance on the first midterm and homework to that date.
Tue 8/281. Joint class: Introduction, motivation: what are real numbers? (Grushevsky)Read pages 1-19
Thu 8/302. Joint class: Properties of numbers; induction; concept of a field. (Berger)HW due 9/5: 1.3, 1.4, 1.10, 1.12, 2.2, 2.5, 3.1, 3.4, 3.6
Tue 9/43. Joint class: Completeness axiom for real numbers; Archimedean property. (Berger)Read pages 20-38
Thu 9/6 4. Joint class: Infinity, unboundedness. Intro to sequences. (Berger)HW due 9/12: parts eghimr of: 4.1,4.2,4.3,4.4; and 4.8,4.10,4.11,4.12,4.14
Tue 9/115. Joint class: Limit of a sequence. (Grushevsky)HW due 9/19: 5.2, 5.6, 7.3, 7.4, 8.1ac
Thu 9/136. Joint class: Limit laws for sequences. (Grushevsky)Read pages 39-55
Tue 9/187. Joint class: Divergence to infinity, more formal proofs. (Berger)HW due 9/26: 8.3, 8.6, 8.8, 8.10, 9.1, 9.3, 9.5, 9.12, 9.14
Thu 9/208. Joint class: Monotone and Cauchy sequences. (Berger)Read pages 56-65
Tue 9/259. Joint class: Subsequences. (Grushevsky)No HW due 10/3 because of the midterm
Thu 9/27Joint Midterm I in Library E4320.Practice midterm 1, Practice midterm 2, Practice midterm 2 solutions
Tue 10/210.Joint class: Subsequences. (Grushevsky)Read pages 66-78 HW due 10/10: 10.1, 10.2, 10.5, 10.8, 10.9, 11.2, 11.4, 11.5, 11.8, 11.9

The following syllabus below is only for MAT 320, in Math P-131
Subject to ongoing changes - please check regularly, and reload the page!
Thu 10/411. Lim sup, lim inf, seriesRead sections 12,14,15,16
Tue 10/9No class, Fall break
Thu 10/1112. Series and series convergence testsHW due 10/17: PDF
Tue 10/1613. Decimal expansionsRead sections 16,17,18
Thu 10/1814. ContinuityHW due 10/24: HW 7
Tue 10/2315. Properties of continuous functions Read sections 18, 20
Thu 10/2516. Limits HW due 10/31: HW 8
Tue 10/3017. Topology in a metric spaceRead section 13
Thu 11/118. Topology in a metric spaceRead section 21
Tue 11/619. Topology in R^nHW due 11/7:HW 9
Tue 11/6Midterm review session by Jack Burkart, MAT P-131, 7pm-8pm
Thu 11/8Midterm 2 Practice midterm, and Another practice midterm
Tue 11/1320. Continuity in a metric space Baire category HW due 11/15 (change!):HW 10
Thu 11/1521. Connectedness and uniform continuityRead sections 22, 19
Tue 11/2022. Uniform continuityshort HW due 11/20: HW 11(a)
Thu 11/22No class - Happy Thanksgiving!
Tue 11/2723.Power seriesRead sections 23,24,25
Thu 11/2924.Uniform convergenceshort HW due 11/28: HW 11(b)
Tue 12/425.Derivative and mean value theoremRead sections 28,29,31
Thu 12/626.Taylor's theoremHW due 12/5: HW 12

Final Exam: Thu December 20, 8.00AM-10.45AM in Math P-131.
Practice final for 320.
Jack's review session Mon December 17, 7.00PM-8.30PM in Math 5-127
Sam's extra office hours in Math 3-109: Mon Dec 17, 10.00-11.30, Tue Dec 18 12.30-1.30

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