**Math 5201- Introduction to Real Analysis- Autumn 2016**

**Instructor**: Krystal Taylor; taylor.2952; MW 706; office hours: MW 11:20 am- 12:20-pm or by apt.

**Recitation leader**: Joseph Migler; migler.1; MW 700; office hours: TR 11:20 am- 12:20-pm or by apt.

**Lecture**: MWF 10:20 am – 11:15 am, Bolz Hall – Room: 432

**Recitation**: TR (9:10 am – 10:05 am, Derby Hall – Room: 060) & (10:20 am – 11:15 am, University Hall – Room: 024 )

**Grading**: quizzes (15%), home work (25 %), midterm (25%), final (35%)

**Text**: W. Rudin, Principles of Mathematical Analysis, 3^{rd} Ed., McGraw-Hill

**Class notes:** We are going to follow the book, but expect some deviations.

**Review sessions:** I am going to run periodic review sessions to take extra questions and to help you prepare for both the midterm and final exams.

**Expectations:** This is a graduate course and I expect you to approach it like professionals. This includes the following:

i) I expect the attendance to be at or near %100.

ii) I expect you to know all the material from the previous lectures throughout the semester. This is necessary to follow the lectures and to do well on quizzes.

iii) I expect you to read the material we are going to cover ahead of time.

iv) I expect all the electronic devices to be off during class and attention focused squarely on the blackboard.

v) I expect you to ask questions in class and suggest alternative approaches to proofs.

**documents**:

Intermediate Value Theorem (see Theorem 3.15 on this sheet for a simple proof of the I.V.T.)

l’Hospital’s: when to use- when not to use

**Homework assignments**:

Set 1: 2, 4, 7, 8, 10, 13 (optional: 15) (due: at the beginning of lecture, Wednesday, Aug. 31) solutions

Set 2 (there are a total of 5 problems): (Rudin Ch. 2) #6, 7, 19; (Bergman) #2.2:6; solutions

Set 3 (there are a total of 6 problems): (Rudin Ch. 2) #12, 16, 17; (Bergman) #2.3:1, 2,3:3 (assume K is infinite); (Problems) X4; (due: at the beginning of lecture,Wednesday, Sept. 14); solutions

Set 4 (there are a total of 5 problems): (Rudin Ch. 2) #26; (Rudin Ch. 3) #1, (Bergman) #2.3:2, (adrians-q), (problems) X24; solutions

Set 5 (there are a total of 10 problems due Oct. 5): (Rudin Ch. 3) 6, 11, 17, 20, 22, 23.(Bergman) #3.2:2, #3.9:0, (problems) X23, X28*. ( solutions I, x23-solution, x28-solution, solutions II)

Set 6 (there are a total of 5 problems due Oct. 20): (Rudin Ch. 4) 1, 4, 8. (Bergman) #3.10:0, 3.13:1. (solutions)

Set 7 (due October 26): (Rudin Ch. 4) #6, 10, 14, 18, 23; (solutions)

Set 8 (there are a total of 5 problems due Nov. 2): (Rudin Ch. 5) #2, 11, 15, 16; (problems) X213 (you may assume that Y is R^k) solutions

Set 9 (there are a total of 6 problems due Nov. 9): (Rudin Ch. 6) #1, 2, 3, 8, 16. (Bergman) 6.3:0 solutions

Set 10 (there are a total of 5 problems due Nov. 16): (Rudin Ch. 6) #10 abc, 12; (Bergman) 6.4:6, 6.3:1; (problem given in recitation Nov. 10) solutions

Set 11 (there are a total of 4 problems due Nov. 22): (Rudin Ch. 7) 9, 11; (Bergman) 7. 1: 1; (Riemann-integration) Exercise 11

Set 12 (there are a total of 6 problems due Dec. 7): (Rudin Ch. 7) 13, 18, 20;(Bergman) 7.1:2; (problems) X172; exercise: Let A and B be dense sets on the real line. Is the intersection of A and B dense?

**Quiz Solutions:**

quiz 1 key

quiz-6-(solutions discussed in class)

quiz-8 (solution in text, solution to bonus discussed in recitation Nov. 10)

**Homework policy: **Homework assignments will appear on each Wednesday of the semester and will due on the following Wednesday at the beginning of lecture (some adjustments around University holidays may announced in class). Late homework creates difficulty for the grader and puts the student behind. For the benefit of all involved, late homework will not be accepted unless extenuating circumstances are present.

**Midterm date: **Wednesday, October 12, ( in class) MIDTERM REVIEW

**Final Exam Date:** Tuesday, December 13, 10-11:45 am

Review for the final (follow W. Rudin, Principles of Mathematical Analysis, 3^{rd} Ed., McGraw-Hill):

- know the definition of the supremum (infimum) of a set, know when this value exists, and be ready to apply
- know elementary properties of the derivative of a function.
- know the definition and properties of the closure of a set (see quiz 2 along with related proofs in Rudin)
- prove Theorem 2.37 and deduce that every bounded sequence has a convergent subsequence
- know the statement of the Heine-Borel Theorem (Theorem 2.41) and be ready to apply
- prove the Intermediate Value Theorem (see Theorem 3.15 on this sheet for a simple proof of the I.V.T.)
- know the statement of the Mean value theorem and be ready to apply
- know Minkowski’s inequality (See HW Set 10 solutions, first page)
- prove Theorem 4.8 (a useful characterization for continuity)
- know when the inverse image of an open set is open, when the image of a compact set is compact, when the extreme value theorem (Theorem 4.16) holds, and demonstrate related counter examples
- statement of Theorem 6.6 (criteria for Riemann integrability) and related definitions
- statement of Theorem 6.17 and how to use it to prove the F.T.C. (See HW Set 10 solutions, 6.5:1)
- statement of Theorem 7.12 (the uniform limit of continuous functions is continuous)
- examine the necessity of the hypotheses of Theorem 7.16
- know the statement of Stone-Weierstrass and how to apply it
- know related counter examples: a differentiable function whose derivative has a discontinuity, a function which is not continuous, but is the limit of continuous functions
- prove the Weirstrass Theorem (every bounded infinite subset of R^d has a limit point in R^d)
- be familiar with concepts presented in class on the last week (we will discuss equicontinous families of functions)
- here are some extra practice problems: practice final problems . You may also look at the problems worked in recitation for practice.

**Supplementary materials**:

**Disability statement**:Students with disabilities that have been certified by Student Life Disabilities Services (SLDS) will be appropriately accommodated and should inform the instructor as soon as possible of their needs. SLDS contact information: slds@osu.edu; 614-292-3307; 098 Baker Hall, 113 W. 12th Avenue; webpage: http://www.ods.ohio-state.edu.

**Academic misconduct statement**: It is the responsibility of the Committee on Academic Misconduct to investigate or establish procedures for the investigation of all reported cases of student academic misconduct. The term “academic misconduct” includes all forms of student academic misconduct wherever committed; illustrated by, but not limited to, cases of plagiarism and dishonest practices in connection with examinations. Instructors shall report all instances of alleged academic misconduct to the committee (Faculty Rule 3335-5-48.7). For additional information, see the Code of Student Conduct at http://studentlife.osu.edu/csc/.