Analysis I MAT 320 MAT 640 Fall 2019

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 page

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

Class Meets: M/W 2:00-3:40 pm in Gillet 305

Office Hours: M/W 1:30-2:00 and 3:40-4:00 pm in Gillet 305

Grading Policy:

Expectations: 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.

Homework: Approximately four hours of homework will be assigned in each lesson as well as additional review assignments over weekends. The reading includes material not covered in class. Note that a single problem may take an hour. In the schedule below, homework is written below the lesson when it is assigned and *ed problems are more important.

Exams: There are three exams(20% each) and a final (40%). If an exam needs to be retaken because it was missed or due to poor performance, the score on the retake will be multiplied by .8 resulting in a maximum score of only 80%. This helps a student avoid failing a course but does not help with a earning an A or a B. The exams consist entirely of proofs.

Materials, Resources and Accommodating Disabilities

Textbook: Mathematical Analysis: a Straightforward Approach by Binmore, 2nd Ed Cambridge University Press .

Available free at https://archive.org/details/MathematicalAnalysis ISBN: 9780521288828

also see Analysis Proofs by Prof Sormani

Materials on Reserve in the Library: There are books about proving techniques available on reserve in the library as well as a complete set of handwritten lecture notes taken by a Lehman College math major.

Tutoring: There is no tutoring for this course, but you may stop by professor's office hours regularly.

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

The schedule will be available on the course webpage listing each lesson’s topics with the homework to be completed after the lesson beneath the lesson. There will be 28 Lessons worth of material. Extra lessons are inserted in case there are class cancellations due to weather or illness later in the semester. If you miss class then you can learn directly from the textbooks or emailed notes.

(Wed 8/28) Lesson 1: Quantifiers

HW: Complete the Quantifiers Worksheet, Read 7.1-7.5, Read 7.6-7.12

No Lesson: (Mon 9/2) Holiday

(Wed 9/4) Lesson 2: Proofs, Counter examples, and Proof by Contradiction

HW: Read 1.1-1.4, rewrite examples 1.5 and 1.6 as two column proofs, do 1.8/exercise *1,*2,*4,

Read the Rules of Proof website

(Th 9/5) Lesson 3: Practice

HW: Read 1.7 and rewrite as a two column proof, do 1.8/ *5, *6, Read 1.9-1.20, 1.12/1*,2,3*,5,6; 1.20/2*, 3*, 6*; 7.16/2*,3*,4*

(Mon 9/9) Lesson 4: Continuoum

HW: Read 2.1-2.9; Do 2.10/1,2*,3,6*; Read 7.9-7.16; Do 7.16/3*,4*; Read the wikipedia page on metric spaces.

(Wed 9/11) Lesson 5: Sup and Inf, Archimedian Property

HW: Read 2.12-2.13; Do 2.13/1*; 7.16/6*; Read 3.1-3.6; Do 3.6/1*,2*

(Mon 9/16) Lesson 6: Convergence on Metric Spaces

HW: Read 4.1-4.5, Do 4.6/1*,2*,3* (be sure to use epsilon in these)

(Wed 9/18) Lesson 7: Convergence and the Sandwich Lemma

HW: Read 4.7-4.9 and go over class notes carefully,

Prove that if a_n converges to A and b_n converges to B then (2a_n + 3 b_n) converges to (2A +3B).

Read 4.10-4.12, Do 4.20/3*

(Mon 9/23) Lesson 8: Practice for Exam I

(Wed 9/25) Lesson 9: Exam I on Sequences will have 4 parts:

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

Any student that scored below 20 on a problem may retake that problem after first submitting both exam’s versions of that problem until it is done perfectly as HW. Resubmissions must begin immediately. Retakes are on 10/14 after class. The new score will be multiplied by .8 so it is a max of 20 even if done perfectly.

(Mon 9/30) Extra Lesson 10: Monotone Sequences

HW: Read 4.14-4.16, Rewrite the proof in 4.17 and examples 4.18 as two column proofs, Read 4.19, Do 4.20/1*,2, 6* Prove that a sequence which is increasing and bounded above converges to its sup. Read the rest of chapter 4, do 4.29/2,4.

(Wed 10/2) Lesson 11: Subsequences, liminf, limsup, the Bolzano Weierstrass Theorem and Cauchy Sequences

HW: Read 5.1-5.7, Read 5.16-5.19 on Cauchy sequences, Do 5.21/1*, 4*

(Mon 10/7) Lesson 12: Limits of Functions

HW: Read photos of todays lecture and do exercises (see email), Read 8.1-8.5, Do 8.15/2*,3*, Prove Prop 8.12 (i) using Defn in 8.3,

(Wed 10/9) No Lesson: Lehman Holiday

(Mon 10/14) Extra Lesson 13: Proof by Induction (and retakes of Exam I after class 4-6 pm)

HW: Read 3.7-3.9, Do 3.11/1i*,1ii*, 2*,3*,4*, Extra Credit due 10/28: Do 1.20/4, 3.6/6,

(Wed 10/16) Lesson 14: Continuity

HW: Consulting any calculus text book prove the squeeze theorem and the three special limits by turning the calc textbook proof into a rigorous proof, Read 8.6-8.7, Do 8.15/ 6*, Read 8.8, Prove Prop 8.12 (ii) and (iii) using Theorem 8.8*, Read 8.13-8.14, Do 8.15/ 5*, Read 8.6, 9.1-9.3 Prove 9.4(i)(ii)(iii) see hint below the three statements, Read 8.16, Prove 9.5* and 9.6*, Practice review for Exam 2 (these have been emailed to the class as well as solutions on Wed night)

(Mon 10/21) Lesson 15: Review for Exam II

Catch up on all homework assigned in Lesson 10-14 before the Exam including reading textbook and photoed notes.

HW do before next Mon: Read 5.1-5.3, Read 5.8-5.10 and then read 5.11-5.14, do 5.15/4,5,6.

(Wed 10/23) Lesson 16: Exam II will have 4 parts:

Part I: a proof by Induction,

Part II: short questions about given sequences using the definitions of bounded, increasing, decreasing, liminf, limsup, Thm 4.17, Thm 4.10, Thm 4.25 and Thm 5.2.

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

(Mon 10/28) Lesson 17: Uniform Continuity, Equicontinuity and limits of functions

HW: Prove the following functions are continuous, check if they are uniformly continuous and equicontinuous:

1) f_n(x)=x/n on [0,1]

2) f_n(x)= nx on [0,1]

3) f_n(x) =xn/(n+1) on [0,1]

4) f_n(x) = x^n on [0,1]

(Wed 10/30) Lesson 18: Uniform Convergence and the Arzela-Ascoli Theorem

HW: Read 9.10, 9.13-9.14, Prove 9.10 imitating the proof in 9.9, Read 9.12 Prove 9.12 for infimum*, Do 9.17/1, 2*,3,4,5,6.

1) f_n(x)=x/n on [0,1]

2) f_n(x)= nx on [0,1]

3) f_n(x) =xn/(n+1) on [0,1]

4) f_n(x) = x^n on [0,1]

Show the first sequence converges uniformly to f(x)=0, the second sequence is unbounded and has no limit, the third sequence converges to f(x)=x uniformly and the last sequence converges pointwise to f(x)=0 for x in [0,1) and f(x)=1 for x=1 but not uniform convergence,

Finally prove the uniform limit of an equicontinuous sequence is continuous.

(Mon 11/4) Lesson 19: Differentiation

HW: Read 10.1-10.3, 10.4-10.10, Do 10.11/2*, Read 10.12-10.14 Do 10.15/2*, 5*

(Wed 11/6) Lesson 20: Mean Value Theorem and Extrema

HW: Read 11.1-11.7, Do 11.8/2*,3*,4*, Read 11.9, Do 11.11/2*,3*

(Mon 11/11) Lesson 21: Area and Integration

Read the "Riemann Sum" article on wikipedia. For 1-4 below: First write the left sum, right sum, upper sum and lower sums for the following five integrals using evenly spaced intervals so that x_i-x_i-1=(b-a)/N (but don't try to evaluate the sum). Which is largest? Which is smallest? Second find the relationship between epsilon and delta for each of these uniformly continuous functions using the mean value theorem (Hint: delta=epsilon/M where M=max|f'|). Finally estimate how large N must be to guarantee an error of epsilon' in each integral. Verify this works. Verify that the upper sum is larger than the lower sum and the left and right sums are in between, and that the upper sum minus the lower sum has an error less than epsilon' when N is chosen large enough.

1) f(x)=4x integrated from 0 to 3

2) f(x)=5x+2 integrated from 3 to 8

3) f(x)=10-2x integrated from 1 to 4

4) f(x)= (x-2)² + 1 integrated from 1 to 3

5) Prove the integral of a constant function f(x)=c from a to b is c(b-a), by taking the sums and explicitly evaluating them.

Workshop for Grad School Applications: Personal Statement:

What to Put In . . . What to Leave out . . . How to Get Started.

Wednesday, Nov. 13, 3:30 - 5:00 PM in Carman Hall, room B04

(Wed 11/13) Lesson 22: Riemann Integral and the Fundamental Thm of Calculus

Read 13.1-13.3, Write proofs for 13.4 and 13.6, Read 13.16-13.17, Show that a function which is 0 on irrational numbers and 1 on rational numbers is not Riemann Integrable, Look over 13.9-13.15, Read 13.19-13.22 carefully, look over 13.23-13.25, Do 13.26/1, 2.

(Mon 11/18) Lesson 23: Series

HW: Review 4.4, 4.10, 4.17, then Read 6.1-6.3, write out proofs of 6.2, 6.3

Prove that the series 1/2 + 1/4 + 1/8 + 1/16+... =1 using proof by induction to verify the partial sums add up to 1-(1/2^k) and using an epsilon N limit proof to show those partial sums converge to 1.

Read 6.4-6.5, Read 6.8, Write out a proof of 6.9,

(Wed 11/20) Lesson 24: Convergence of Series

HW: Read 6.10-6.11, Write out a proof of 6.11 Review Cauchy sequences in 5.16, 5.17, 5.19,

Read 6.12-6.15 Rewrite the proofs of 6.16-6.19 (in 6.18 you may assume the limit is 0 rather than the limsup).

(Mon 11/25) Lesson 25: Taylor Series and Convergence of Functions

HW: Review Calculus II textbook section on Taylor Series about x=0 and steps taught in class including the induction proof to:

1) find the Taylor series for e^(3x) and check where it converges using the ratio test.

2) find the Taylor series for 1/(1-x) and check where it converges using the ratio test.

3) Find the Taylor series for Ln(x+1) and check where it converges.

4) Prove f_n(x)=xⁿ defined on [0,1] converges pointwise to a function which is 0 everywhere on [0,1) and is 1 at 1.

Does this last function converge uniformly? Prove or disprove.

(Wed 11/27) Lesson 26: Review for Exam III

A sample exam will be distributed that students can work on together. Students may also ask questions about

the homeworks assigned corresponding to each of the four parts of the exam.

(Mon 12/2) Lesson 27: Exam III on the Proofs of Calculus I will have 5 problems:

Problem I: Prove that a sequence of functions is equicontinuous as in Lesson 17

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 21 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 25

Problem V: Check where the Taylor Series converges using a test as in Lesson 25

(Wed 12/4) Lesson 28: Review of Calculus I - II from an advanced perspective

Students meet without the professor to complete the following assignment in teams (do not divide up the work: do it together)

1) Prove the chain rule using the definition of derivative (can read this proof in a calculus textbook)

2) Prove limit of sin(x)/x is 1 and limit of (cos(x)-1)/x is 0 using trigonometry and areas of triangles as in a calculus textbook but adding justifications from Analysis.

3) Prove that the derivative of sine is cosine and the derivative of cosine is sine.

4) Prove u substitution by writing out the chain rule and integrating both sides of the equality.

5) Prove integration by parts, by first noting the product rule (uv)' = u'v + u v' and then integrating both sides of this equality.

(Mon 12/9) Lesson 29: Review of Riemann Integration, Equicontinuity, Mean Value Theorem, Uniform Convergence, and Arzela-Ascoli Theorem, (Retake Ex 3 Prob 1 or EC Quiz on Equicontinuity in class)

HW: Review Lessons 23, 24, and 25 and practice Taylor series in a Calculus Textbook

(Wed 12/11) Lesson 30: Review of Sups, Infs, Limits, Riemann Integration, Improper Integrals and Logs (Retake Ex 3 Prob 4 after class)

HW: Review Lesson 21, 22, Read 8.18-8.19, Do 8.20/1*, 2*, Read 13.27-13.33, Do 13.34/1*, Read 14.1-14.2, Do 14.3/1*, Read 14.4, Do 14.5/1*, Read 14.6, Do 14.7/3*,5*

Retakes of entire Exam II will be Mon 12/16 6:15-8:15,

Retakes of Exam III Prob 2, Prob 3, and Prob 5 are just the corresponding problems on the final.

Review for the Final

You 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..

The final will be in the following form:

• write a complete epsilon-delta proof of continuity or of a limit of a function (as in Exam 2 Part 3 or Part 4)

• write a complete proof that a sequence of functions converges uniformly (as in Exam 3 Prob 2)

• prove an upper bound is a sup using proof by contradiction and Archimedean Principle (as in Exam I Part 2)

• find a Riemann Sum approximating a Riemann Integral up to a given error (as in Exam 3 Part 3)

• find the radius and interval of convergence for a given Power Series (as in Exam 3 Part 5)

Finals Week: Final Exam is on Wed Dec 18 1:30-3:30 pm