CRN: 23367

Time and Place: TTh 10:30–11:45am in SCHM 309

Instructor: Arshak Petrosyan

Office Hours: TTh 9:15–10:30am, 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 \(M\)-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. Homework will be graded for completeness only and feedback will be provided on selected problems. 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). The exact information will be posted on the Exams page at least two weeks in advance.

Grading: You homework and exam scores will be available in Gradescope. Your final score will be calculated by the scheme

	Final Score = 0.4 FE + 0.28 ME1 + 0.28 ME2+ 0.04 HW,

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

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 4: Last day to withdraw from a course without it appearing on record
  • Tue, Nov 24: Last day to withdraw from a course with a W (advisor approval required)

For other important dates see Purdue Registration Calendars.

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.

Collaboration, Use of Online Resources, and AI Tools: Collaboration and the use of outside resources can support learning when done responsibly. Homework is an opportunity to develop the independent understanding required for quizzes and exams, where no outside aids are permitted.

All submitted work must be written in the student’s own words. Any assistance from classmates, textbooks, online resources, or AI tools and large language models (such as ChatGPT or similar systems) must be fully disclosed, including the specific sources and the type of help received. Submitting work that is not one’s own without disclosure will result in zero points on the assignment and may lead to an academic misconduct report.

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 encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247, as soon as possible.

If the Disability Resource Center (DRC) has determined reasonable accommodations that you would like to utilize in this class, you must send your Course Accommodation Letter to the instructor. Instructions on sharing your Course Accommodation Letter can be found by visiting: How To Use Your Course Accommodation Letter. Additionally, you are strongly encouraged to contact the instructor as soon as possible to discuss implementation of your accommodations.

In this course, we will follow Department of Mathematics’ Accommodation Procedures for Students with Disabilities.

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. For cases that fall under the University’s 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 medical care. Absences outside of those covered by the University’s excused class absence regulations are at the instructor’s discretion. Purdue expects each student to be responsible for class-related work missed due to an unavoidable absence. Students should contact their instructors directly to discuss the absence and opportunity to complete missed coursework. This work may be made up at the discretion of the instructor.

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.

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 own potential. The University believes that intellectual and cultural 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.

See Purdue’s Information on Emergency Preparation and Planning. This website covers topics such as Severe Weather Guidance, Emergency Plans, and a place to sign up for the Emergency Warning Notification System. I encourage you to download and review the Emergency Procedures Guide.

Please also review Building Emergency Plan for SCHM (where our classroom is located).