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	MW 11:45-12:40 R03@Javits 110. Online access
TA	Ben Wu and Tobias Shin	Daniel Brogan
Office hours	Before tryig to access office hours via Zoom, Sign into Zoom using SB username. ShengYuan Zhao: Thursday 1-2pm and Friday 11-12 (Zoom) Ben Wu: Mon,Wed 2-3pm (Zoom) Tobias Shin: Fri 4-6pm (in person)	Before tryig to access office hours via Zoom, Sign into Zoom using SB username. Raanan Schul: Tuesday 11:30-12:30 and Thursday 11:30-12:30 (Zoom) Dan Brogan: Mon, Wed 1-2pm (Zoom) and in person
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	For 319, see here. For MAT320, it will be the week of November 2.
Final	Thursday, Dec. 17, 8:00am-10:45am. Online. See BB announcements for more details!

• Blackboard in particular, take a look at
  • Announcements and
  • Discussion Board
  • Zoom lectures
  • Zoom recitations
• The MLC is a great place to get help. It will be fully online in Fall 2020.
• Undergrad advising
• SB "Coming Back Guide"
• Online Self-Screening Tool
• Face Mask Policy

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. 24-28	Tuesday	Sets, Induction (Schul)	Read 1.1-1.3, Appendix A and appendix B
		Thursday	Infinite 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. 4	Tuesday	Algebraic 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.	Tuesday	suprema, completeness, Archimedean property (Zhao)	Read: 2.3,2.4,2.5 until page 48	No recitation Monday, Sep. 7
		Thursday	intervals, decimals (Zhao)	Due Sept. 18: 2.4: 3, 7, 11, 19 2.5: 2, 5, 14 Submitted via BB
4	Sep. 14-18	Tuesday	Decimals. Limits (3.1). [Schul]	Read 3.1, 3.2
		Thursday	Limit Theorems (3.2). [Zhao]	Due Sept 25: 3.1: 5bc, 9, 18 3.2: 1bd, 6ac, 7, 10b submitted via BB
5	Sep 21-25	Tuesday	Finish 3.2. (Zhao)	Read: 3.2, 3.3, 3.4	Midterm Monday, Sept 21. Includes weeks 1-4
		Thursday	3.3, start 3.4 (Schul)	Due Oct. 2nd: 3.3: 1,6,9. submitted via BB Read example 3.3.6

Starting week 6, MAT319 and 320 are not held together

The following is for MAT320

Week#		Day	Topic	Homework	Note
6	Sep. 28-Oct. 2	Tuesday	3.4 subsequences and BW theorem	Read 3.4, 3.5, 3.6
		Thursday	3.5 Cauchy criterion	Due Oct. 9: 3.4: 11, 16, 19 3.5: 4, 6, 10 submitted via BB
7	Oct. 5-9	Tuesday	3.6: Divergent sequences. 3.7: Infinite Series.	Read 3.6, 3.7, 4.1, 4.2
		Thursday	4.1: Limits of functions. 4.2: Limit theorems	Due Oct. 16: 3.6: 9 3.7: 2, 5, 11, 12, 17 4.1: 8, 12c, 14, 15 4.2: 5
8	Oct. 12-16.	Tuesday	5.1+5.2:Cont. Func. and combinations	Read 5.1, 5.2, 5.3
		Thursday	5.3: Cont. Func. on Intervals	Due Oct. 23: 5.1: 1, 3, 11 5.2: 7, 10 5.3: 4, 6, 17
9	Oct. 19-23	Tuesday	5.4 Uniform continuity	Read 5.4, 5.6
		Thursday	finish 5.4 and also 5.6: Monotone functions.	Due Oct. 30: 5.4: 8, 9, 14 5.6: 7, 10, 12
10	Oct. 26-30	Tuesday	finish 5.6 + 6.1	read 6.1-6.3
		Thursday	6.2+6.3	Due Nov. 6: 6.1: 3, 8c, 12 6.2: 12, 14, 15
11	Nov. 2-6	Tuesday	6,3, 6.4	Read 6.3, 6.4, 7.1	Midterm 2 this week. AND Elections. Go vote.
		Thursday	6.4	Due Nov. 13 6.3: 1,3,4 6.4: 13, 17, 18
12	Nov. 9-13	Tuesday	7.1, 7.2	Read 7.1, 7.2, 7.3
		Thursday	7.2	Due Nov. 20 7.1: 4, 8, 9, 15 7.2: 7, 8, 12, 13
13	Nov. 16-20	Tuesday	7.3	Read 7.3, 7.4, 8.1, 8.2
		Thursday	8.1, 8.2	Due Wednesday, Dec. 2 7.3: 5, 9, 15, 20
14	Nov. 23-27	Tuesday			No classes this week. Happy Thanksgiving
		Thursday
15	Nov. 30- Dec. 4	Tuesday	8.2, 8.3		Fully online this week
		Thursday	9	Due Monday, Dec. 7 8.1: 4, 14, 19, 23

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