Time and Place: MWF 11:30am–12:20pm in UNIV 103

Instructor: Arshak Petrosyan

Office Hours: MWF 10:30am-11:30pm, or by appointment, in MATH 836

Course Description: Credit Hours: 3.00. Real analysis in one and n-dimensional Euclidean spaces. Topics include the completeness property of real numbers, topology of Euclidean spaces, Heine-Borel theorem, convergence of sequences and series in Euclidean spaces, limit superior and limit inferior, Bolzano-Weierstrass theorem, continuity, uniform continuity, limits and uniform convergence of functions, Riemann or Riemann-Stieltjes integrals.

Learning Outcomes: 1. Perform rigorous proofs using the definitions of open sets, closed sets, connected sets, compact sets, interior points, boundary points, cluster points, finite sets, infinite sets, and denumerable sets in Euclidean spaces. 2. Perform rigorous proofs of convergence or divergence for sequences or series in Euclidean spaces. 3. Determine points of continuity and existence of limits using rigorous proofs for functions whose domain and range are in Euclidean spaces. 4. Perform rigorous proofs that a sequence of functions converges uniformly or does not converge uniformly on a subset of a Euclidean space. 5. Know and be able to apply the definition and related theorems on the existence of Riemann or Riemann-Stieltjes integrals.

Textbook:

[B] R. Bartle, The Elements of Real Analysis, Second Edition, John Wiley & Sons, New York, 1975.

Course Outline:

  • The algebraic, ordering, and completeness properties of the real numbers. (3 hrs.)
  • Topology of Rp. (5 hrs.)
  • Sequences in Rp. Convergence and Uniform convergence. Lim Sup and Lim Inf. (5 hrs.)
  • Continuous and uniformly continuous functions. sequences of continuous functions. Approximation Theorems. (7 hrs.)
  • Differentiation. Mean Value and Taylor’s Theorem. (3 hrs.)
  • The Riemann (Riemann-Stieltjes) Integral. Improper Integrals. (6 hrs.)
  • Infinite Series of constant and functions. Absolute and Uniform Convergence. Weierstrass -Test; Dirichlet and Abel Test. Power Series. Double Series and the Cauchy Product. (8 hrs.)

Homework: There will be weekly homework assignments to be collected through Gradescope, typically due at 11:59pm on Wed or Fri. No late homeworks will be accepted, however the lowest homework score will be dropped. For more information, see Homework page.

Exams: There will be two midterm exams (evening exams) and a comprehensive final exam (covering all material).

  • Midterm Exam 1: Thu 10/06, 2022 8:00p - 9:30p LILY G420
  • Midterm Exam 2: Thu 11/17, 2022 8:00p - 9:30p LILY G420
  • Final Exam: Week of 12/12–12/17 (to be scheduled)

More details will be posted on the Exams page.

Grading: Your final grade will be computed by the scheme

Final Score = (1/3)FE + (7/30)ME1 + (7/30)ME2 + (1/5)HW,

where where FE, MEi, HW are the scores (in %) for Final Exam, Midterm i, Homework.

Note: If you perform better than average on both midterm exams, you will be given an option of not taking the final exam and your score will be computed by an alternative scheme (to be specified towards the end of the course).

Grade cutoffs: Students who get at least 97% of the total points in this course are guaranteed an A+, 93% guarantees an A, 90% an A-, 87% a B+, 83% a B, 80% a B-, 77% a C+, 73% a C, 70% a C-, 67% a D+, 63% a D, and 60% a D-; for each of these grades, it’s possible that at the end of the semester a lower percentage will be enough to get that grade.

Important dates

  • Fri, Sep 2: Last day to cancel a course assignment without it appearing on record
  • Tue, Oct 25: Last day to withdraw from a course with a W or WF grade

Academic Integrity: As a reminder, all students must comply with Purdue’s policy for academic integrity:

https://www.purdue.edu/odos/osrr/academic-integrity/

Academic integrity is one of the highest values that Purdue University holds. Individuals are encouraged to alert university officials to potential breeches of this value by either emailing integrity@purdue.edu or by calling 765-494-8778. While information may be submitted anonymously, the more information that is submitted provides the greatest opportunity for the university to investigate the concern.

Students with Disabilities: Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are welcome to let me know so that we can discuss options. You are also encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone at 765-494-1247.

If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact your instructor to discuss your accommodations as soon as possible. Here are instructions for sending your Course Accessibility Letter to your instructor:

https://www.purdue.edu/drc/students/course-accessibility-letter.php

Attendance Policy: This course follows Purdue’s academic regulations regarding attendance, which states that students are expected to be present for every meeting of the classes in which they are enrolled. When conflicts or absences can be anticipated, such as for many University-sponsored activities and religious observations, the student should inform the instructor of the situation as far in advance as possible. For unanticipated or emergency absences when advance notification to the instructor is not possible, the student should contact the instructor as soon as possible by email or phone. When the student is unable to make direct contact with the instructor and is unable to leave word with the instructor’s department because of circumstances beyond the student’s control, and in cases falling under excused absence regulations, the student or the student’s representative should contact or go to the Office of the Dean of Students (ODOS) website to complete appropriate forms for instructor notification. Under academic regulations, excused absences may be granted by ODOS for cases of grief/bereavement, military service, jury duty, parenting leave, or emergent or urgent medical care.

Classroom Guidance Regarding Protect Purdue: Any student who has substantial reason to believe that another person is threatening the safety of others by not complying with Protect Purdue protocols is encouraged to report the behavior to and discuss the next steps with their instructor. Students also have the option of reporting the behavior to the Office of the Student Rights and Responsibilities. See also Purdue University Bill of Student Rights and the Violent Behavior Policy under University Resources in Brightspace.

Academic Guidance in the Event a Student is Quarantined/Isolated: If you must miss class at any point in time during the semester, please reach out to me via Purdue email so that we can communicate about how you can maintain your academic progress. For COVID-19 concerns, please see the Fall 2022: What you need to know guidance published July 27. If you find yourself too sick to progress in the course, notify your adviser and notify me via email or Brightspace. We will make arrangements based on your particular situation.

Mental Health/Wellness Statement: Purdue University is committed to advancing the mental health and well-being of its students. If you or someone you know is feeling overwhelmed, depressed, and/or in need of mental health support, services are available. For help, such individuals should contact Counseling and Psychological Services (CAPS) at 765-494-6995 during and after hours, on weekends and holidays, or by going to the CAPS office on the second floor of the Purdue University Student Health Center (PUSH) during business hours. The CAPS website also offers resources specific to situations such as COVID-19.

Nondiscrimination Statement: Purdue University is committed to maintaining a community which recognizes and values the inherent worth and dignity of every person; fosters tolerance, sensitivity, understanding, and mutual respect among its members; and encourages each individual to strive to reach his or her potential. In pursuit of its goal of academic excellence, the University seeks to develop and nurture diversity. The University believes that diversity among its many members strengthens the institution, stimulates creativity, promotes the exchange of ideas, and enriches campus life. For more information, please see Purdue’s full Nondiscrimination Policy Statement.

Emergencies: In the event of a major campus emergency, course requirements, deadlines and grading percentages are subject to changes that may be necessitated by a revised semester calendar or other circumstances beyond the instructor’s control. Relevant changes to this course will be posted onto the course website or can be obtained by contacting the instructor via email or phone. You are expected to read your @purdue.edu email on a frequent basis.