Analysis I MAT 320 MAT 640 Fall 2017

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

Course Webpage: https://sites.google.com/site/professorsormani/teaching/AnalysisI-S16

Professor Sormani: google "Sormani Math" or go to http://comet.lehman.cuny.edu/sormani

Contact: sormanic@gmail.com (do not call the office and leave messages) Office Hours: 1:30-2:00, 5-6:00 M/W in Gillet 200A

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

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.

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

Consult the course webpage for updated homework assignments.

Lesson 1 (Mon 8/28): Quantifiers HW: Complete the Quantifiers Worksheet, Read 7.1-7.5, Read 7.6-7.12

Lesson 2 (Wed 8/30): Proofs and Counter examples 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

No classes at Lehman on Mon 9/4

Lesson 3 (Wed 9/6): Proof by Contradiction 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*

Lesson 4 (Mon 9/11): Continuoum HW: Read 2.1-2.9; Do 2.10/1,2*,3,6*; Read 7.9-7.16; Do 7.16/3*,4*

Lesson 5 (Wed 9/13): 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*

Lesson 6 (Mon 9/18): Convergence HW: Read 4.1-4.5, Do 4.6/1*,2*,3* (be sure to use epsilon in these)

No classes at Lehman on Wed 9/20 (students may meet and study together)

Lesson 7 (Mon 9/25): Convergence and the Sandwich Lemma HW: Read 4.7-4.9 and go over class notes carefully, Prove that if an converges to A and bn converges to B then (2an + 3 bn) converges to (2A +3B). Read 4.10-4.12, Do 4.20/3*

Lesson 8 (Wed 9/27): Practice for Exam I (solutions have been emailed to the class, email me if you did not receive them)

Review notes for Exam I (which is described below) are at the bottom of the page and were emailed to the class.

Lesson 9 (Mon 10/2): 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

Lesson 10 (Wed 10/4): Proof by Induction

HW: Read 3.7-3.9, Do 3.11/1i*,1ii*, 2*,3*,4*, Extra Credit due 10/25: Do 1.20/4, 3.6/6,

No classes at Lehman on Mon 10/9

Exam I Continued at Home: Do all exercises on the Exam I Review Sheet and submit on 10/11

(your grade will be averaged with the in class Exam I grade) The sheet is at the bottom of this page.

Lesson 11 (Wed 10/11): 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.

Lesson 12 (Mon 10/16): 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*

Lesson 13 (Wed 10/18): Limits of Functions 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,

Lesson 14 (Mon 10/23): 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*,

Lesson 15 (Wed 10/25): 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 Wed: Read 5.1-5.3, Read 5.8-5.10 and then read 5.11-5.14, do 5.15/4,5,6.

Lesson 16 (Mon 10/30): 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

Lesson 17 (Wed 11/1): Uniform Continuity, Equicontinuity 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.

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]

Then 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.

For students told to redo parts of Exam II, the redone parts will be averaged with the original grades on those parts and the total will be recomputed. The redone parts must be redone during office hours sometime in the next two weeks.

Lesson 18 (Mon 11/6): 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*

Lesson 19: (Wed 11/8) 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*

Lesson 20 (Mon 11/13): 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 xi-xi-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.

Lesson 21 (Wed 11/15): 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.

Submit your solutions to the HW from Lesson 20 for feedback by Monday 11/20

Lesson 22 (Mon 11/20): Limits, Improper Integrals and Logs

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*

Exam III is a take home exam and is posted below under Wed lesson due next Monday.

Lesson 23 (Wed 11/22): Review of Calculus I from an advanced perspective

HW: 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) (too hard) Prove that e^{x+y}=e^x e^y by first taking f(x)= e^{x+y}-e^x e^y, then showing f(0)=0, then showing f'(x)=0 everywhere, then concluding f(x)= const=0 everywhere.

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

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

Exam III on the Proofs of Calculus I will have 4 parts and is a Take Home Exam due Mon 11/27

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

f_n(x)=(2nx)/(n+4) on [5,7] (you may consult your notes but not work together nor ask for help)

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

show the sequence from part I converges uniformly to f(x)=2x on [5,7] (you may consult your notes but not work together nor ask for help)

Part 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

f(x)=1/x on [1,5] and get an error less than .0001 (you may consult your notes but not work together nor ask for help)

Extra credit if you actually use a computer to find the value of the sum up to this error. Maple is recommended and available at Lehman.

Part IV: HW from Lesson 23

Do five of the problems, worth five points each, (you may consult calc textbooks and online material being sure to cite your source)

Lesson 24 (Mon 11/27): 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,

Lesson 25 (Wed 11/29): Convergence of Series

At 6 pm in 333: Guest Speaker at Lehman:

Dr. Ndiaye presents Finding Minima and Barycenters

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).

Lesson 26 (Mon 12/4): Taylor Series and Convergence of Functions

HW: Review Calculus 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 fn(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 function converge uniformly? Prove or disprove.

Lesson 27 (Wed 12/6): 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 and functions, the Taylor series for a given function and its radius of convergence.

  • The final will be in the following form:

  • write a complete proof that a function is continuous

  • write a complete proof that a sequence converges.

  • write the first line of a proof by contradiction,

  • start a proof by induction: base case and first few lines of the proof including the application of the induction hypothesis

  • find a Riemann Sum approximating a Riemann Integral up to a given error

  • find limits, sups and infs, limsups and liminfs without proof

  • find taylor series and verify convergence of the series

Lesson 28 (Mon 12/11) Final Exam 2 pm - 5 pm

Finals Week: (Wed Dec 20 1:30-3:30 pm in 333) Extra Credit Exam or Retake/Makeup Final

There will be an extra difficult extra credit exam for top students who wish to do something impressive that could be described in a letter of recommendation for graduate school. A retake of the final exam will also be given during finals week for those who need it. The Extra Credit Exam is only for students who do not need to retake the final.