CRN 31912

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  • Course: Math 3341 (Introduction to Analysis) CRN 31912 - M-F 9:20-11:30 in BELL 130A
  • Instructor: Dr. Helmut Knaust, Bell Hall 124, tel. 747-7002, e-mail:
  • Office Hours: M-F 11:30-12:20, or by appointment
  • Textbook: There is no textbook. Class notes will be provided by the instructor.
  • Prerequisites: The course requires a certain level of mathematical maturity that you have gained by having thoroughly and successfully grappled with the concept of infinity in MATH 1312 as well as the basics of logic and proofs in MATH 3325.
  • Course Objectives: Real Analysis is "Calculus with Proofs". You should expect (and I will expect) that you make considerable progress in the following areas:
  1. Become familiar with the fundamental results of "Analysis on the Real Line" (highlights of the course include the Intermediate Value Theorem, and possibly the Mean Value Theorem and the Fundamental Theorem of Calculus);
  2. Thoroughly understand the definitions of the basic concepts of Analysis such as convergence, continuity, differentiation and integration;
  3. Continue to develop your ability to use the method of proof to establish these fundamental results.
  4. Be able to recognize a rigorous proof when you read or see one. Conversely, be able to pick out the weak spot(s) in a less rigorous argument. Be able to fill in details in a sketchy proof.
  5. Once you have devised a proof, be able to write it down in a clear, concise manner using correct English and mathematical grammar.
  6. Be able to present and defend a proof to a group of your peers.
  • In-class Activities and Presentations: Mathematics is not a spectator sport. Therefore I do not give lectures. I will call on students to give presentations of exercises in the class notes. I will also regularly ask for volunteers to present solutions to tasks at the blackboard. Your presentations are the most important part of the course. Your chances of passing the course without spending a significant amount of time on preparing in-class demonstrations are zero: Your in-class work, evaluated for both quality and quantity, will account for 30% of your grade.
  • Tests: Two 60-minute exams will be given on Tuesday, July 3 and Friday, July 13. Each exam counts 20% of your grade.
  • Final Examination: The final exam on Monday, July 23, at 10:00-12:45, is comprehensive and mandatory. It counts 30% of your grade.
  • Graduate Credit: If you take this undergraduate course for graduate credit, you will additionally write up solutions to Optional Tasks 2.2-2.6, 2.9-2.14, and 3.6-3.8. This assignment is due on Friday, July 20, and will be graded for an additional 20% of your grade. (Since your possible score therefore increases to 120%, I will divide your total score by a factor of 1.2 to get to a 100% scale.)
  • Time Requirement: I expect that you spend an absolute minimum of four to five hours a day outside of class on reading the class notes, preparing for the next class, reviewing your own notes, and completing assignments. Not surprisingly, it has been my experience that there is a strong correlation between class grade and study time.
  • Attendance: Due to the course structure, attendance is mandatory. An unexcused absence will result in an exercise/task grade of 0 for the day of the absence. Four absences (excused or unexcused) will lead to dismissal from the class with a grade of "F".

Ground Rules

  • The notes distributed in class contain “exercises" and “tasks". Students will solve these problems at home and then present the solutions in class. The instructor will call on students at random to present “exercises"; he will call on volunteers to present solutions to the “tasks".
  • When in the audience, students are expected to be actively engaged in the presentation. This means checking to see if every step of the presentation is clear and convincing, and speaking up when it is not. When there are gaps in the reasoning, the students in class will work together to fill the gaps.
  • The instructor serves as a moderator. His major contribution in class will consist of asking guiding and probing questions. He will also occasionally give short presentations to put topics into a wider context, or to briefly talk about additional concepts not dealt with in the notes.
  • Students may use only the class notes and their own notes taken during the semester; they are not allowed to consult other books or materials. Students must not talk about assignments to anyone other than class participants and the instructor. Students are encouraged to collaborate with other class participants; if they do, they must acknowledge other students’ contributions during their presentation. Exemptions from these restrictions require prior approval by the instructor.
  • The instructor is an important resource. He expects frequent visits from all students in class during his office hours – many more visits than in a “normal" class. Among other things, students probably will want to come to the instructor’s office to ask questions about concepts and assigned problems, they will probably occasionally want to show the instructor their work before presenting it in class, and they probably will have times when they just want to talk about the frustrations they may experience.
  • It is of paramount importance that we all agree to create a class atmosphere that is supportive and non-threatening to all participants. Disparaging remarks will be tolerated neither from students nor from the instructor.

Class Progress

  • 7/20/07: 5.11-5.17
  • 7/19/07: 4.12, 5.1, 5.4-5.7
  • 7/18/07: 4.16, 4.19, 4.20, 5.2, 5.3
  • 7/17/07: 4.17, 4.18
  • 7/16/07: 4.13-4.15
  • 7/13/07: Test 2; 4.8-4.10
  • 7/12/07: 3.8, 4.1-4.7
  • 7/11/07: 3.2-3.7
  • 7/10/07: 2.33\(\Leftarrow\), 2.34, 2.35, 3.1
  • 7/9/07: 2.18\(\Rightarrow\), 2.27, 2.31-2.32, 2.33\(\Rightarrow\)
  • 7/6/07: 2.26, 2.28-2.31 (2)
  • 7/5/07: 2.14, 2.21-2.25
  • 7/3/07: Test 1, 2.19, 2.20
  • 7/2/07: 2.15-2.17, 2.18\(\Leftarrow\)
  • 6/29/07: 2.10-2.13
  • 6/28/07: 1.10, 2.4-2.9
  • 6/27/07: 1.8, 1.11, 2.1-2.3
  • 6/25/07 - 6/26/07: 1.1, 1.3.-1.7, 1.9

Open Problems

  • None.


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