from the course catalogue: “Careful development of elementary real analysis for those who intend to take graduate courses in Mathematics. Topics include completeness property of the real number system; basic topological properties of n-dimensional space; convergence of numerical sequences and series of functions; properties of continuous functions; and basic theorems concerning differentiation and Riemann integration.“
who, when, where
lecturer: Yuliy Baryshnikov.
lecture: Monday-Wednesday-Friday, 10-10:50am, David Kinley Hall 322.
office hours: Thursday, 2-3pm, CSL163.
interactions: campuswire, gradescope (code GVB53X).
schedule (lectures, topics, homework)
- Aug 25 Monday: Course overview. Natural Numbers. Peano Axioms, Mathematical Induction. Notes
- Aug 27 Wednesday Rational and Irrational Numbers. Fields.
- Aug 29 Friday Algebraic numbers. Ordered fields. Notes. Homework 1 released on gradescope.
- Sep 3 Wednesday Real Numbers. The Completeness Axiom. Notes.
- Sep 5 Friday Class Test 1
- Sep 8 Monday Countability. Cardinal numbers. Diagonal Trick. Notes.
- Sep 10 Wednesday Limits of Sequences. Formal proofs. Notes.
- Sep 12 Friday Limit Theorems for Sequences. Class is prerecorded, – here.
- Sep 15 Monday. Limit Theorems for Sequences, cont’d. Notes.
- Sep 17 Wednesday Monotone Sequences. Notes.
- Sep 19 Friday Class Test 2
- Sep 22 Monday \(\limsup\)’s and $\liminf$’s. Notes.
- Sep 24 Wednesday Cauchy Sequences Notes.
- Sep 26 Friday Subsequences, subsequential limits. Notes.
- Sep 29 Monday Some Topological Concepts in Metric Spaces. Notes.
- Oct 1 Wednesday More Topological Concepts in Metric Spaces. Notes.
- Oct 3 Friday Class Test 3
- Oct 6 Monday Series. Notes.
- Oct 8 Wednesday Decimal Expansions of Real Numbers. Continuous Functions. Notes.
- Oct 10 Friday Continuous Functions. Notes.
- Oct 13 Monday Continuous Functions on Compacts. Notes.
- Oct 15 Wednesday Uniform Continuity. Limits of Functions. Notes.
- Oct 17 Friday Class Test 4
- Oct 20 Monday More on Metric Spaces: Continuity Notes.
- Oct 22 Wednesday More on Metric Spaces: Compactness Notes.
- Oct 24 Friday More on Metric Spaces: Connectedness. Notes.
- Oct 27 Monday Power Series. Notes.
- Oct 29 Wednesday Uniform Convergence. Notes.
- Oct 31 Friday Class Test 5
- Nov 3 Monday The Tale of Two Pancakes.
- Nov 5 Wednesday More on Uniform Convergence. Notes.
- Nov 7 Friday Differentiation and Integration of Power Series. Notes.
- Nov 10 Monday Abel’s Theorem. Bernstein polynomials.
- Nov 12 Wednesday Basic Properties of the Derivative
- Nov 14 Friday Class Test 6
- Nov 17 Monday More on Derivatives Notes
- Nov 19 Wednesday The Mean Value Theorem Notes
- Nov 21 Friday The Riemann Integral Notes
- Dec 1 Monday Properties of the Riemann Integral Notes
- Dec 3 Wednesday Fundamental Theorem of Calculus
- Dec 5 Friday Class Test 7
- Dec 8 Monday
- Dec 10 Wednesday Survey
syllabus
- text: Kenneth Ross, Elementary Analysis, 2nd ed., available through library as an e-text.
- homework will be assigned biweekly, but not graded. (ignore it at your own peril.)
- grades will be based on 5 best of the 7 class tests, ~35 minutes each, see schedule above (5*15%=75%), and an oral final exam (25%).
statements
Academic Integrity
The University of Illinois at Urbana-Champaign Student Code should also be considered as a part of this syllabus. Students should pay particular attention to Article 1, Part 4: Academic Integrity. Read the Code at the following URL: http://studentcode.illinois.edu/.
Academic dishonesty may result in a failing grade. Every student is expected to review and abide by the Academic Integrity Policy: https://studentcode.illinois.edu/article1/part4/1-401/. Ignorance is not an excuse for any academic dishonesty. It is your responsibility to read this policy to avoid any misunderstanding. Do not hesitate to ask the instructor(s) if you are ever in doubt about what constitutes plagiarism, cheating, or any other breach of academic integrity.
Emergency Response Recommendations
Emergency response recommendations can be found at the following website: http://police.illinois.edu/emergency-preparedness/. I encourage you to review this website and the campus building floor plans website within the first 10 days of class. http://police.illinois.edu/emergency-preparedness/building-emergency-action-plans/.
Sexual Misconduct Reporting Obligation
The University of Illinois is committed to combating sexual misconduct. Faculty and staff members are required to report any instances of sexual misconduct to the University’s Title IX Office. In turn, an individual with the Title IX Office will provide information about rights and options, including accommodations, support services, the campus disciplinary process, and law enforcement options.
A list of the designated University employees who, as counselors, confidential advisors, and medical professionals, do not have this reporting responsibility and can maintain confidentiality, can be found here: wecare.illinois.edu/resources/students/#confidential.
Other information about resources and reporting is available here: wecare.illinois.edu.
Religious Observances
Illinois law requires the University to reasonably accommodate its students’ religious beliefs, observances, and practices in regard to admissions, class attendance, and the scheduling of examinations and work requirements. You should examine this syllabus at the beginning of the semester for potential conflicts between course deadlines and any of your religious observances. If a conflict exists, you should notify your instructor of the conflict and follow the procedure at https://odos.illinois.edu/community-of-care/resources/students/religious-observances/ to request appropriate accommodations. This should be done in the first two weeks of classes.
Disability-Related Accommodations
To obtain disability-related academic adjustments and/or auxiliary aids, students with disabilities must contact the course instructor and the Disability Resources and Educational Services (DRES) as soon as possible. To contact DRES, you may visit 1207 S. Oak St., Champaign, call 333-4603, e-mail disability@illinois.edu or go to https://www.disability.illinois.edu. If you are concerned you have a disability-related condition that is impacting your academic progress, there are academic screening appointments available that can help diagnosis a previously undiagnosed disability. You may access these by visiting the DRES website and selecting “Request an Academic Screening” at the bottom of the page.
Family Educational Rights and Privacy Act (FERPA)
Any student who has suppressed their directory information pursuant to Family Educational Rights and Privacy Act (FERPA) should self-identify to the instructor to ensure protection of the privacy of their attendance in this course. See https://registrar.illinois.edu/academic-records/ferpa/ for more information on FERPA.
Public Health
Student location address You should insert and/or update your “Student Location” in Student Self-Service every time your living location changes. Typically, this is at the beginning of each academic year but could change throughout the year.
Maintaining updated location information enables campus health and wellness units to communicate proper protocols to students that will help us all maximize safety during the pandemic.
Feeling ill and absences: Students need to take responsibility for checking their symptoms every day. Students who feel ill must not come to class. These students are judged to have excused absences for the class period and should contact the instructor via email about making up the work.
Run > Hide > Fight
Emergencies can happen anywhere and at any time. It is important that we take a minute to prepare for a situation in which our safety or even our lives could depend on our ability to react quickly. When we’re faced with almost any kind of emergency – like severe weather or if someone is trying to hurt you – we have three options: Run, hide or fight.
Run: Leaving the area quickly is the best option if it is safe to do so. Take time now to learn the different ways to leave your building. Leave personal items behind. Assist those who need help, but consider whether doing so puts yourself at risk. Alert authorities of the emergency when it is safe to do so.
Hide: When you can’t or don’t want to run, take shelter indoors. Take time now to learn different ways to seek shelter in your building. If severe weather is imminent, go to the nearest indoor storm refuge area. If someone is trying to hurt you and you can’t evacuate, get to a place where you can’t be seen, lock or barricade your area if possible, silence your phone, don’t make any noise and don’t come out until you receive an Illini-Alert indicating it is safe to do so.
Fight: As a last resort, you may need to fight to increase your chances of survival. Think about what kind of common items are in your area which you can use to defend yourself. Team up with others to fight if the situation allows. Mentally prepare yourself – you may be in a fight for your life.
Please be aware of people with disabilities who may need additional assistance in emergency situations.
Other resources:
police.illinois.edu/safe for more information on how to prepare for emergencies, including how to run, hide or fight and building floor plans that can show you safe areas.
emergency.illinois.edu to sign up for Illini-Alert text messages.
