Fall 2020 MAT 319: Foundations of Analysis Fall 2020 MAT 320: Introduction to Analysis Schedule TuTh 9:45-11:05am Online TuTh 9:45-11:05am Online Instructor ShengYuan Zhao Raanan Schul Recitation MW 11.45-12.40: R01@Online; R02@Frey 102; R03@Javits 110 MW 11:45-12:40 @ TBA TA Ben Wu and Tobias Shin and Daniel Brogan Daniel Brogan Office hours Before tryig to access office hours via Zoom, Sign into Zoom using SB username. Note: in the first 5 weeks, you can go to any of the above office hours, no matter what recitation you are in 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 MAT 250 or permission of instructor; C or higher in one of the following: MAT 203, 211, 220, 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 B or higher in MAT 200 or MAT 250 or permission of instructor; C or higher in one of the following: MAT 203, 211, 220, 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 Textbook Introduction to Real Analysis 4th Edition by Bartle and Sherbert, 4th edition Homework Weekly problem sets will be assigned. 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. Your lowest homework grade will be dropped at the end of the class. Homework will be submitted via Blackboard Grading Homework: 20%, Midterm I: 20%, Midterm II: 20%, Final: 40%. Midterm I Will be during recitation, in person, Monday Sept 21. Exceptions to the "in person" policy will only be given for documented extenuating circumstances. Midterm II TBA Final Thursday, Dec. 17, 8:00am-10:45am. Online

Syllabus/schedule (subject to change) The first 10 lectures will be joint between MAT319 and MAT320. After that, we will provide each you you with a reccomendation as to whether you should go on to take MAT319 or MAT320. These will be based on your performance in the homework and first midterm. The plan is that the material in the first 4 weeks will cover the first 2 chapters and part of the 3rd chapter. The first midterm will be on the 5th week during recitation (in person).

Joint MAT319/320 classes
Week# Day Topic Homework Note
1 Aug. 24-28Tuesday Sets, Induction (Schul) Read 1.1-1.3, Appendix A and appendix B
ThursdayInfinite sets (Zhao)Due Sep. 4:
• 1.1:7, 10, 14, 22
• 1.2: 14, 20
• 1.3: 2 (also complete the proof of part (a) of Theorem 1.3.4), 9, 12
Submitted via Blackboard
2 Aug. 31 - Sep. 4TuesdayAlgebraic properties of the real numbers + inequalities (Schul) Read 2.1-2.3
Thursday Abs. Value + Completeness (Schul)Due Sept. 11:
• 2.1:2, 4, 20
• 2.2: 5,17
• 2.3: 4, 14
Submitted via Blackboard
3 Sep. 7-11.Tuesdaysuprema, completeness, Archimedean property (Zhao)Read: 2.3,2.4,2.5 until page 48No recitation Monday, Sep. 7
Thursdayintervals, decimals (Zhao)Due Sept. 18:
• 2.4: 3, 7, 11, 19
• 2.5: 2, 5, 14
Submitted via BB
4 Sep. 14-18TuesdayDecimals. Limits (3.1). [Schul] Read 3.1, 3.2
ThursdayLimit Theorems (3.2). [Zhao] Due Sept 25:
• 3.1: 5bc, 9, 18
• 3.2: 1bd, 6ac, 7, 10b
submitted via BB
5 Sep 21-25TuesdayFinish 3.2. (Zhao) Read: 3.2, 3.3, 3.4 Midterm Monday, Sept 21. Includes weeks 1-4
Thursday3.3, start 3.4 (Schul) Due Oct. 2nd: 3.3: 1,6,9. submitted via BB

Starting week 6, MAT319 and 320 are not held together

The following is for MAT319

For MAT320 see here
Week# Day Topic Homework Note
6 Sep. 28-Oct. 2Tuesday Bolzano-Weierstrass Theorem Read 3.4, 3.5
Thursday Cauchy sequences Due Oct. 9:
• 3.4: 6, 9, 14
• 3.5: 4, 10, 12
submitted via BB
7 Oct. 5-9 Tuesday Divergent sequence and series Read 3.6, 3.7
Thursday Series Due Oct. 16:
• 3.6: 1
• 3.7: 3b, 4, 10
submitted via BB
8 Oct. 12-16.TuesdayLimits of functions and limit theorems Read 4.1, 4.2, 5.1, 5.2
Thursday Continuous functions and combinations of continuous functions Due Oct. 23:
• 4.1 : 6, 9a, 9d,
• 5.1: 6, 8, 10
submitted via BB
9 Oct. 19-23TuesdayContinuous functions on intervals Read 5.3, 5.4 until p143
Thursday Continuous functions on intervals and uniform continuity Due Oct. 30:
• 5.3 : 1,2,4
• 5.4: 2,6
submitted via BB
Thursday Monotone and inverse functions No HW this week
11 Nov. 2-6Tuesday Finish 5.6, Derivatives Read 6.1Midterm 2 Monday 11/02
ThursdayFinish 6.1, begin 6.2 Due Nov. 13:
• 6.1 : 5a,5c, 9, 10
submitted via BB
12 Nov. 9-13Tuesday6.2 Read 6.2,6.3 untill 177
Thursday6.3 Due Nov. 20:
• 6.2: 5,6,13,19
• 6.3: 1,6
submitted via BB
13 Nov. 16-20Tuesday 7.1 Riemann integral Read 7.1, 7.2
Thursday7.2 Integrable functions Due Dec. 4:
• 7.1: 2,3,6a,7,8,10
• 7.2: 8,10,15
submitted via BB
14 Nov. 23-27Tuesday No classes this week. Happy Thanksgiving
Thursday
15 Nov. 30- Dec. 4Tuesday 7.2 Integrable functions Read 7.2, 7.3Fully online this week
Thursday7.3 The fundamental theorem Due Dec. 9:
• 7.3:2,6,10,15,18ab
submitted via BB

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