The text for the course is Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger.
Your grade will be based on several homework assignments (20%), one Midterm (20%), the Writing in Major assignment (20%) and a Final exam (40%).
This course satisfies the Writing in Major requirement. In addition to learning the material, we shall focus on writing clear and complete mathematical proofs. In your homework assignments, you will therefore be required to write in complete, grammatical sentences; you will be graded both with respect to mathematical accuracy and for clarity of expression. If you find writing proofs difficult, please do talk to Henry or me during office hours.
About two thirds of the way through the course there will be a writing assignment due. I will give details of this project later. The writing project is an integral part of the course, and it applies also to non-math majors.
Students who elect (or change their election later) to Credit/No Credit should first check with me so that they have a clear idea of how much work is expected for a passing grade.
We will have an in class (closed book, closed notes) exam on April 22 (Thursday). The midterm will cover material from chapters 1-26 of the book.
Midterm problems My solutions The midterm didn't go so well for some of you; please be sure to study carefully the solutions, and to clarify any difficulties you have with Henry or me during office hours. Here is the grade distribution for the mid term: 90-83 A (3 students), 78-70 A- (8 students), 67-59 B+ (7 students), 56-51 B (4 students), 46 B- (1 student), 42-37 C+ (4 students), 34-29 C (2 students), 24-22 C- (2 students), 17-14 D+ (2 students).
Timeline for the WIM project: The first three parts of the project should be straight-forward based on what we have covered so far. You should turn in a first draft covering these topics by Tuesday May 18. You should turn in a draft of topics 4, 5 and 6 by Tuesday May 25. Our WIM grader is Kaveh Fouladgar and he will give you feedback on your drafts. The complete project is due on the last day of class June 1.
The final exam will be in class, closed book and closed notes. It will take place on Friday, June 4 from 3:30-6:30PM in Room 380X. The final exam will cover material from chapters 1-46, 48-50, 60, 66-67 of the book.
Homework 1: Due Thursday April 8. Exercises: 2.2, 2.3, 3.3, 3.5, 4.4, 4.7, 5.1, 5.7, 6.3, 6.4, 7.5, 7.10, 8.3, 8.5. Solutions thanks to our CA Henry.
Homework 2: Due Thursday April 15: Exercises: 9.6, 9.11, 10.12, 11.8, 11.12, 12.4, 12.7, 13.1, 13.3, 15.4, 16.5, 16.13, 16.15, 18.1, 18.5. Solutions thanks to our CA Henry.
Homework 3: Due Thursday April 22. Exercises: 17.4, 19.3, 20.6, 20.7, 20.9, 20.13, 20.20, 21.2, 22.4, 23.5, 24.9, 25.2, 25.4, 26.4, 26.8. Solutions thanks to our CA Henry.
Homework 4: Due Thursday April 29. Exercises: 26.5, 27.1, 27.2, 28.1, 28.3, 28.6, 29.3, 29.5. Solutions thanks to our CA Henry.
Homework 5: Due Thursday May 6. Exercises: 29.15, 35.3, 35.7, 35.8, 35.9, 36.4, 36.8, 36.11, 37.4, 37.9, 37.10. Solutions thanks to our CA Henry.
Homework 6: Due Thursday May 13. Exercises: 38.6, 38.8, 38.9, 38.14, 39.4, 39.5, 39.7, 39.9, 40.8, 40.10, 40.15, 40.17, 41.3, 41.4. Solutions thanks to our CA Henry.
Homework 7: Due Thursday May 20. Exercises: 42.1, 42.3, 42.5, 42.10, 42.12, 43.1, 43.4, 43.7, 44.1, 44.6 (a,b,c), 44.8. Solutions thanks to our CA Henry.
Homework 8: Due Thursday May 27. Exercises: 44.2, 44.5, 44.7, 45.1, 45.2, 45.4, 46.1, 46.4, 46.7, 46.8, 60.2, 60.5. Solutions thanks to our CA Henry.
March 30: Properties of real numbers: Read Chapters 1-5 in book.
April 1: Integers, rationals, countable sets: Read Chapters 6-8 in book.
April 6: More on countable and uncountable sets: Chapters 8 and 9 of the book.April 8: Sequences of real numbers. Convergent, bounded, monotone. Bolzano-Weierstrass theorem. Read Chapters 10-16 and 18.
April 13: Cauchy sequences. Lim sup and lim inf. Infinite series. Read Chapters 17, 19-22 of book.
April 15: Absolute convergence of series. Alternating series test. Harmonic series. Ratio and root tests. Chapters 23-26 of book.
April 20: Rearrangement of terms: absolutely convergent ok, not ok for conditionally convergent series. Power series. Double series. Read Chapters 27 to 29. Especially go through Chapter 28 which we didn't cover in class, but which you should learn on your own.
April 22: In class Mid-term.
April 27: Double series and product of series. Metric spaces definitions and examples. Chapters 29, 35 and 36 of book.
April 29: Cauchy-Schwarz, more examples of metric spaces, convergence of sequences in a metric space. Chapters 35-37 of book.
May 4: Open and closed sets. Continuous functions on metric spaces. Chapters 38-40. Read chapters 30-33 also; that should be largely review of continuous functions on R.
May 6: Continuous functions on metric spaces. Relative metric. Beginning of compactness. Read chapters 40 and 41.
May 11: Compactness and Sequential compactness. Equivalence in metric spaces. Heine-Borel theorem for R. Read chapters 42 and 43 of book.
May 13: Continuous functions on compact spaces. Connectedness. Read chapters 43-45 of book.
May 18: Uniform continuity of continuous functions on compact metric spaces. Connectedness of R^n, intermediate value theorem. Beginning of discussion of complete metric spaces. Read chapters 44-46.
May 20: Discussion of complete metric spaces. Example of continuous functions on [0,1] with uniform metric. R^n, l^1. Read Chapters 46 and 60.
May 25: Completing a metric space. Finish reading chapter 46.
May 27: Discussion of differentiation. Mean-value theorem, Taylor's theorem. Exponential and logarithmic functions. Read chapters 48-50, and 66-67.
June 1: The Weierstrass approximation theorem.