Analysis I

e-mail: laurentiu.maxim@lehman.cuny.edu
office hours: Thu, 2-6 PM, Room G 100B.

CLASS SCHEDULE: Tue, Thu, 7:45-9:25 PM, Room G 305.

CLASS HOMEPAGE: http://comet.lehman.cuny.edu/maxim/320.html

DESCRIPTION:

Introduction to Real Analysis; the real number system, limits, continuity, the mean value and Taylor's theorems and applications, Riemann integration and improper integrals.

TEXTBOOK:

Mathematical Analysis: a straightforward approach
K.G. Binmore, 2nd Edition, Cambridge University Press.

PREREQUISITE: MAT 226 or equivalent vector calculus.

FINAL EXAM: May 22, 8-10 PM, Room G 305.

GRADE: 10 Quizzes: 5% each, Midterm: 20%, Final Exam: 30%.

THE QUIZES COUNT MORE TOWARDS THE GRADE THAN EITHER OF THE EXAMS! Students must submit completed homework from the prior week in order to begin a quiz and that homework will be part of the score. Students who miss quizes because they are absent, late or have not completed the homework will receive a 0 on that quiz. There will be no make up quizes.

Note that a quiz may contain proofs and the proofs should be written in a two column format with statements and justifications. Very similar proofs will be on the homework that is required to take the quiz. If you are unsure of the proof of a problem, you may write a question mark next to the step you are unsure of.

In the following syllabus, homework is written below the lesson when it is assigned and should be completed before the next meeting. All quizes and homework collection are on Tuesdays.

SYLLABUS (updated regularly):

• Lesson 1, Tue 1/29: 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
• Lesson 2, Thu 1/31: Quantifiers, Roots, Irrational numbers
  HW: Read 1.9-1.20, do 1.8/exercise 5, 6; do 1.12/exercise 1, 3, 5; do 1.20/exercise 1, 3, 6.
• Lesson 3, Tue 2/5: Quiz 1 + more on Roots, Modulus and Functions.
  HW: Read 7.1-7.15, do 7.16/exercise 2, 3, 4.
• Lesson 4, Thu 2/7: Continuum property
  HW: Read 2.1-2.9; Do 2.10/ex: 1, 2, 3, 6. Read 2.12-2.13; Do 2.13/ex: 1, 3.
• Lesson 5, Thu 2/14: Quiz #2, Principle of Induction
  HW: Read 3.1-3.10; Do 3.6/ex: 1, 2 and 3.11/ex: 1, 2.
• Lesson 6, Tue 2/19: More on "Natural numbers; Principle of Induction"
  HW: Do 3.6/ex: 3, 4, 5 and 3.11/ex: 4.
• Lesson 7, Thu 2/21: Convergence
  HW: Read 4.1-4.5, Do 4.6/ex: 1, 2 ,3 (be sure to use epsilon in these)
• Lesson 8, Tue 2/26: Quiz 3 (on induction) and more on Convergence
  HW: Read 4.7-4.10, Do 4.20/ex: 1, 2. Read 4.11-4.14.
• Lesson 9, Thu 2/28: Monotone sequences
  HW: Rewrite the proof in 4.17 and examples 4.18 as two column proofs, Read 4.19, Do 4.20/ex: 3, 6.
• Lesson 10, Tue 3/4: Quiz 4 (on convergent sequences) + Subsequences.
  HW: Read 4.21-4.28. Do 4.29/ex: 3, 4, 5. Read 5.1-5.6. Do 5.7/ex: 1, 2, 3.
• Lesson 11, Tue 3/6: Bolzano Weierstrass Theorem, Cauchy sequences
  HW: 5.21/ex: 1,4. Prepare questions for the review session.
• Lesson 12, Tue 3/11: Review for the midterm exam
  HW: Go over class notes and quizzes 1-4.
• Lesson 13, Tue 3/13: MIDTERM EXAM
• Lesson 14, Tue 3/18: Discussion on solutions to the MIDTERM EXAM problems.
• Lesson 15, Tue 3/20: Series.
  HW: Read 6.1-6.11 and work out more examples from your old Calc II textbook. Also familiarize yourself with the convergence tests for series (again from Calc II).
• Lesson 16, Tue 3/25: Convergence Tests for Series.
  HW: Read the proofs of the convergence tests and work out as many examples as you can from 6.26 and from your old Calc II textbook.
• Lesson 17, Tue 3/27: Quiz #5 on Convergence Tests for Series.
• Lesson 18, Tue 4/1: Quiz #6 + Limits of Functions.
  HW: Read 8.1-8.5, Do 8.15/ex. 2, 3 , Prove Prop 8.12 (i) using Defn in 8.3.
• Lesson 19, Tue 4/3: Continuity of Functions.
  HW: Read 8.6-8.14, Do 8.15/ex. 4, 5, 6.
• Lesson 20, Tue 4/8: Quiz #7 and more on Continuity.
  HW: Do 9.17/ex 1, 2, 3, 6.
• Lesson 21, Tue 4/10: Differentiation
• Lesson 22, Tue 4/15: Rules of Differentiation.
  HW: Do 10.11/ex 1, 2, 3. Do 10.15/ex 4, 6.
• Lesson 23, Tue 4/17: Mean Value Theorems + Quiz #8.
  HW: Do 11.8/ex 1, 2, 4, 5. Do 11.11/ex 1, 2.
• Lesson 24, Tue 4/29: Monotone Functions.
  HW: Read 12.1-12.8, Do 12.12/ex 1, 2, 3. Read 12.13-12.20, Do 12.21/ex 1.
• Lesson 25, Tue 5/1: Integration.
  HW: Read 13.1-13.3 Write proofs for 13.4,13.5,13.6,13.7, Read 13.8-13.15, Study 13.11 and 13.15 closely.
• Lesson 26, Tue 5/6: Riemann Integral.
  HW: Read 13.16, Do 3.17, Do 3.18/ ex. 1, 2. Read 13.20-13.25, Do 13.26/ex. 1, 2, 4.
• Lesson 27, Tue 5/8: Quiz #10 and Review for the Final Exam.
  
• Lesson 28, Tue 5/13: Quiz #11 and Review for the Final Exam.