Analysis I MAT 320 MAT 640 Spring 2021

MAT320 Analysis I: 4 hours, 4 credits. Introduction to real analysis, the real number system, limits, continuity, differentiation, the mean value theorem, Taylor's theorems and applications. Riemann integration and improper integrals.

Prerequisite: Either Vector Calculus MAT226 or Departmental permission

Professor Sormani: google "Sormani Math Lehman” for my webpage

Contact: sormanic@gmail.com (do not call the office and leave messages)

Class Meets: Asynchronously Online. Students complete and submit each lesson when they wish, keeping in mind that there are 28 lessons in the course that must be completed before taking the final exam. Students who fall behind may risk failing the course. Students cannot request an INC unless they have passed three exams during the semester.

Office Hours: Email questions in an email with subject line MAT320 Question and the professor will respond within 48 hours. Questions about homework must include a photo of the statement of the problem and your initial attempt.

Expectations: This course involves writing proofs and all proofs must be completed in two column format with numbered statements and justifications as taught in this course. Students are expected to learn both the mathematics covered in class and the mathematics in the textbook and other assigned reading. Completing homework is part of the learning experience. Students should review topics from prior courses as needed using old notes and books.

Classwork: Classwork is required as proof of attendance. All your work before Exam 1 should be photographed and the photos will be pasted into a googledoc entitled MAT320-1-Lastname-Firstname with your last name and your first name the googledoc will be shared with sormanic@gmail.com as an editor. Note that you can easily do this on a smart phone using the googledoc app by copying your photos and then pasting into the googledoc.

Homework: In addition to the classwork, approximately four hours of homework will be assigned in each lesson that should be completed before starting the next lesson. Note that a single problem may take an hour. In the schedule below, homework is written within the lesson notes. Classwork and *ed homework must be submitted as proof of completion of each lesson in the same googledoc.

Exams: Exams will be given on Tuesdays 3-5pm or 9-11pm. Each exam may only be taken two days after all the lessons leading to the exam have been completed and submitted. There are no retakes, so do not take an exam unless you are both healthy and prepared. All students will have individual exams and must upload each part of their exam to their googledocs within 25 minutes. There are three exams (20% each) and a final (40%).

Materials, Resources and Accommodating Disabilities

Lecture Notes: Will be provided under each lesson

A Textbook: (not required) Mathematical Analysis: a Straightforward Approach by Binmore, 2nd Ed Cambridge University Press ISBN: 9780521288828 (was free but no longer, is not required)

Supplementary Notes: Analysis Proofs by Prof Sormani

Calculus Textbook: you should have a copy of your calculus textbook in case you do not remember everything from that course and need to review it. MIT Free Calc Textbook

Accommodating Disabilities: Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need classroom accommodations are encouraged to register with the Office of Student Disability Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441.

Accommodating Holidays: If you have a holiday during a lesson or extra lesson, let me know, and something will be arranged for you.

Names/Gender: We will use last surnames in this class. You may call me Sormani or Professor.

Respect: All students will treat each other with respect and dignity. Let me know if you have concerns.

Course Objectives

At the end of the course students should be able to:

1. find limits, sups and infs by applying theorems (as part of department objectives in math A, B & E)

2. prove that a sequence converges and a function is continuous at a point (as part of E, F & G)

3. write a proof by contradiction (as part of F & G)

4. state, apply and prove theorems related to Calculus including Riemann sums (as part of E)

5. write a proof by induction involving series (as part of F & G)

6. find Taylor series, prove convergence theorems and find radii of convergence (as part of B, E & F)

These objectives will be assessed on the final exam along with other important techniques.

Course Calendar (2 lessons per week starting Feb 1)

Each lesson below will be linked to a googledoc which has links to videos and to classnotes. Classwork must be submitted as well as *ed homework to prove a lesson is completed.

Lesson 1: Introduction to Sets, Direct Proofs, Inequalities, Cases, and Quantifiers (Feb 1-2)

Lesson 2: Bounds, Max/Min, and Proof by Contradiction (Feb 3-4)

Lesson 3: Sup and Inf with the Continuoum Property of the Reals (Feb 8-10)

Lesson 4: Sup and Inf with the Archimedian Property (Feb 11-12)

Lesson 5: Metric Spaces, Balls, Open Sets and Converging Sequences of Points (Feb 15-16)

Lesson 6: Converging and Diverging Sequences in the Real Line (Feb 17-18)

Submit Lessons 1-6 by 11pm Sun Feb 21 to take the exam on Tuesday Feb 23

Review for Exam 1 should have been its own Lesson instead of part of Lesson 6. It was too much for many of the students so Lesson 10 is now a catch up date for students taking the Exam 1 late.

Lesson 7: Exam I on Sequences

will have 4 parts (25 minutes each) aim to take on Tu Feb 23

Part I: an upper bound proof,

Part II: a sup/inf proof,

Part III: an epsilon-N convergence proof for a specific sequence,

Part IV: an epsilon-N convergence proof about sums/differences/products of sequences

Lesson 8: Convergence and the Sandwich Lemma (Feb 24-25)

Lesson 9: Monotone Sequences (Mar 1-2)

Lesson 10: Catch up date for students behind schedule taking the exam this week.

Lesson 11: Subsequences, liminf, limsup, and Bolzano Weierstrass Theorem (Mar 8-9)

Lesson 12: Limits of Functions: lim as x to c of f(x) is L (Mar 10-11)

Lesson 13: Continuity of Functions and Existence of Maxima (Mar 15-16)

Lesson 14: Review for Exam II (Mar 17-18)

Lesson 15: Exam II on Sequences, Continuity, and Limits will have 4 parts (25 minutes each) aim to take on Tu Mar 23. If you complete Lesson 14 by Wed Mar 24 you may take Exam 2 Thursday Mar 25 at 9pm or Friday Mar 26 at 3pm, otherwise take the exam after Spring Break with the usual rule of submission of work by Sunday at 10 pm to take the exam on Tuesday at 3pm or 9pm.

Parts I-II: short questions about given sequences: bounded, increasing, decreasing, liminf, limsup, etc

Part III: an epsilon-delta continuity or limit proof for a specific function,

Part IV: an epsilon delta continuity or limit proof for a combination of functions

Lesson 16: Uniform Continuity and Equicontinuity (Mar 24-25)

Spring Break (Prof Sormani will be away)

Lesson 17: Uniform Convergence, C([0,1]), and the Arzela-Ascoli Theorem (Apr 5-6)

Lesson 18: Differentiation (Apr 7-8)

Lesson 19: Extrema, Mean Value Theorem, Continuity, and Inc/Dec (Apr 12-13)

Lesson 20: Riemann Sums and Integrals (Apr 14-15)

Lesson 21: Riemann Integration and the Fundamental Thm of Calculus (Apr 19-20)

Lesson 22: Series (Apr 21-22)

Lesson 23: Convergence of Series (Apr 26-27)

Lesson 24: Convergence of Taylor Series and Review for Exam 3 (Apr 28-29)

Lesson 25: Exam III on the Proofs of Calculus I - II

will have 4 parts (25 minutes each) aim to take on Tu May 4 (submit work by Sun May 2)

Problem I: Prove that a sequence of functions is equicontinuous as in Lesson 16

Problem II: Prove that a sequence of functions converges uniformly to a limit function as in Lesson 17-18

Problem III: Find the Riemann Sum as in HW from Lesson 20 being sure to state how large N must be taken to have an error less than a given value

Problem IV: Find the Taylor Series as in Lesson 24 and check where it converges using a test as in Lesson 23.

Lesson 26: Natural Log and L’hopital’s Rule (May 5-7)

Lesson 27: Limits and Improper Integrals (May 10-11)

Lesson 28: Limits and Review for the Final Exam (May 12-13)

Final Exam: The final will be given during finals week (May 19-25)

Choose Fri May 21 3pm or 9pm or Sun May 23 10pm or Tu May 25 3pm or 9pm

Students will need to know the statements of all the important theorems and definitions we've learned this semester including sup, inf, bound, limit, bounded increasing sequences converge, sandwich lemma, subsequences of bounded sequences converge, Cauchy sequences, theorems about these, continuity, theorems about this, differentiation, mean value theorem, Rolle's theorem, Riemann integration of continuous functions, theorems about integration, improper integrals, series, convergence tests including comparison, ratio, root and alternating series tests, Taylor series, radius of convergence, uniform convergence. You will also need working knowledge of these concepts in the sense that you must be able to find the limit of various given sequences, the sup and inf of various sets in order to complete the proofs.

Since this course covered so much material it is recommended to spend two days reviewing before the final.

The final will have four parts (25 minutes each):

Part 1: write a complete epsilon-delta or epsilon-R proof of a limit of a function (or of continuity)

Part 2: write a complete proof that a sequence of functions converges uniformly

Part 3: prove an upper bound is a sup using proof by contradiction and the Archimedean Principle

Part 4: find a Riemann Sum approximating a Riemann Integral up to a given error

and find the radius and interval of convergence for a given Power Series