Catalog Description. CRN 23367. 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.
Overview. This is an advanced undergraduate course in proof based mathematics. The course will be divided into 3 roughly equal parts: (I) the real numbers, basic point-set topology of metric spaces, continuity; (II) integration and differentiation; (III) spaces of functions.
Meeting Time and Location. MWF, 4:30-5:20 pm in UNIV 129.
Textbook. The required textbook for this class is "Real Mathematical Analysis," Second Edition, by Charles Chapman Pugh. This is available to download for free as a pdf from Purdue Libraries: See the course Brightspace page under "Content/Textbook".
Though not required, I highly recommend Richard Hammack's "Book of Proof" as a supplementary guide to the basics of mathematical proof. This is also accessible under "Content/Textbook".
Contact. The best way to contact me is by email. My email address is tsincla(at)purdue.edu. I will usually respond to emails fairly promptly during normal business hours. If you have not received a reply within 48 hours, feel free to follow up.
Office. My office is 744 in the Mathematical Sciences building.
Office Hours. See the course Brightspace page for details. Outside of regular office hours, I am also happy to meet by appointment. Please email me to schedule.
Brightspace. The course page on Brightspace will serve as the main point of contact for announcements, assignments, course policy, and virtual content. Students will be expected to check the course page regularly, at least before every class meeting.
Gradescope. Assignments must be uploaded to Gradescope via the course page in Brightspace. No hard copies of assignments will be accepted.
Kaltura. Audio and screen captures of each lecture will be available via Kaltura on the course Brightspace page. Please allow up to 24 hours for processing.
Academic Calendar For ease of reference, here is a link to the academic calendar detailing all breaks, add/drop deadlines, etc.
Lectures. Lectures are held in-person, but written notes and audio will be recorded and posted to Brightspace. Students are strongly encouraged to attend lecture. Class cancellations and other changes will be communicated via the course page in Brightspace.
Accessibility. 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: https://www.purdue.edu/drc/students/course-accommodation-letter.php. Additionally, you are strongly encouraged to contact the instructor as soon as possible to discuss implementation of your accommodations.
Exams. There will two in-class exams during the semester.
Alternate Examinations. Alternately, you may obtain credit on any or all exams by oral examination. Intent to take such an exam must be communicated at least 3 business days prior to the scheduled exam date. Exam scores obtained by oral examination are final and are not subject to dispute. Makeup exams will not be granted without a legitimate, documented excuse. Makeup exams will be oral.
Homework. There will be 8 homework sets. (See below for details and due dates.) Students are encouraged to collaborate on homeworks as long as: 1) all collaborators are listed at the top of each assignment; 2) each student turns in their own, individual work. Rote copying of solutions from peers, use of internet forums, use of AI to assist in completion of homework problems, or plagiarism of any kind will not be tolerated. Any assignment determined by my personal judgement more likely than not to have been completed with the use of prohibited resources or in a prohibited manner will be given a score of zero.
Homework Formatting. Homeworks must be typed or neatly written and scanned. There is a strong preference for assignments to be prepared using LaTeX. (See the section on LaTeX below.) There should be no significant cross-outs, rewrites, scratchwork, scribbling, etc. Problems should be clearly indicated and be placed in the correct sequence. Homework should be uploaded to Gradescope by 11:59 pm on the due date.
Late Homework. No penalty will be assessed for assignments which are not excessively late (less than one week past due).
Semester Project and Presentation. Students will be expected to complete a semester project. The project must be related to, and make use of, mathematical concepts discussed in class and reflect roughly 15 hours of effort per individual. Each team must craft a 1 page proposal briefly describing the project along with a concrete plan for meeting the objectives which must meet my approval. The initial draft of the proposal is due by the end of eighth week. More details will be made available in the "Semester Project" section in the Brightspace course page. Each team will give a 20-minute presentation during the last week of instruction or during the scheduled final examination period if additional time is needed.
Grades. There will be 100 total points. Homework will be worth 5 points per assignment (40 points total). Exams will be worth 20 points each (40 points total) and the semester project will be worth 20 points. For marginal cases, there will be some discretionary leeway in final grade assignment to account for course participation/engagement or extraordinary effort. The following is a sample cutoff distribution which is fairly typical for courses I have previously taught. The final grade cut-offs may differ slightly. A >92, A- >88, B+ >84, B >77, B- >73, C+>69, C> 64, C- >60. 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-.
LaTeX is the language for mathematical typesetting. If you are a CS, Math, or Stats major, I would strongly recommend becoming proficient in LaTeX. Here is the link to A.J. Hildebrand's excellent collection of beginner LaTeX resources. Another great place to start is Jon Peterson's advice and resources for new researchers. You will probably also frequently need to consult the LaTeX Wiki.
Academic Integrity. See the Academic Integrity webpage from the Office of the Dean of Students. Penalties for academic dishonesty will be, at minimum, a score of zero on the exam or assignment. Egregious cases will be referred to the Dean of Students and may result in failure of the course or expulsion.
The following is a tentative outline of topics covered and is subject to change. If you are absent from class it is your responsibility to find out what material was covered and to obtain notes from classmates. Section numbers corresponding to each topic are written parenthetically.
Week 1, 8/25 Basic Set Theory, Logic, and Proof Techniques (1.1) (see also, Book of Proof, chapters 1, 2, and 4-6.)
Week 2, 9/1 LABOR DAY. The Real numbers (1.2), Euclidean space (1.3), Cardinality (1.3)
Week 3, 9/8 Metric spaces (2.1), Continuity (2.2) Completeness (2.3)
Week 4, 9/15 Compactness and Uniform Continuity (2.4)
Week 5, 9/22 Connectedness (2.5), Coverings (2.7), Cantor Sets (2.8)
Week 6, 9/29 Differentiation and Taylor's Theorem (3.1)
Week 7, 10/6 The Riemann Integral (3.2)
Week 8, 10/13 FALL BREAK. The Darboux Integral and Riemann integrability (3.2)
Week 9, 10/20 Series and Series of Functions (3.3). LEEWAY
Week 10, 10/27 Functions Spaces and Uniform Convergence (4.1)
Week 11, 11/3 Power Series (4.2), Analytic Functions (4.6)
Week 12, 11/10 Stone--Weierstrass Theorem (4.4), Banach Contraction Principle (4.5)
Week 13, 11/17 Equicontinuity (4.3)
Week 14, 11/24 LEEWAY. THANKSGIVING BREAK.
Week 15, 12/1 LEEWAY/SPECIAL TOPICS.
Week 16, 12/8 Semester Project Presentations.
Week 1, 8/25 HW 1 Assigned.
Week 2, 9/1 HW 1 Due. HW 2 Assigned.
Week 3, 9/8 HW 2 Due. HW 3 Assigned
Week 4, 9/15 HW 3 Due. Schedule a Preliminary Consultation on Semester Project.
Week 5, 9/22 HW 4 Assigned.
Week 6, 9/29 HW 4 Due.
Week 7, 10/6 EXAM 1.
Week 8, 10/13 DRAFT OF SEMESTER PROJECT PROPOSAL DUE.
Week 9, 10/20 HW 5 Assigned.
Week 10, 10/27 HW 5 Due. HW 6 Assigned.
Week 11, 11/3 HW 6 Due. HW 7 Assigned.
Week 12, 11/10 HW 7 Assigned. Schedule a Consultation on Semester Project.
Week 13, 11/17 HW 7 Due. EXAM 2.
Week 14, 11/24 THANKSGIVING BREAK.
Week 15, 12/1 HW 8 Assigned
Week 16, 12/8 HW 8 Due.