**Professor:**
Roman Vershynin

Office: 3064 East Hall

E-mail: romanv "at" umich "dot" edu

**Class meets:** MWF 10:10-11:00 am (Section 1); 11:10-12:00 noon (Section 2), in 4096 EH.

**Office hours:** Monday, Wednesday, Thursday 2:10-3:00 pm in 3064 EH.
Please do come to my office hours: this is an effective way to
improve your command of the material. You can also grab me after the class if you
have a quick question. I won't be able to hold office hours at any other times.

**Prerequisites:**
A thorough understanding of Calculus and one of 217, 312, 412.

**Course Description:**
This course has two complementary goals: (1) a rigorous development of the fundamental ideas of Calculus,
and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs.
The key words here are **rigor** and **proof**; almost all of the material of the course consists
in understanding and constructing definitions, theorems (propositions, lemmas, etc.), and proofs.
This is considered one of the more difficult among the undergraduate mathematics courses,
and students should be prepared to make a **strong commitment** to the course. In particular,
it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451.

**Textbook** (optional): Robert G. Bartle, Donald R. Sherbert, Introduction to real analysis.
Wiley 4th ed. 2011. Other editions are OK.

**Homework, Exams and Grading:**
There will be two in-class midterm exams and one final exam.
You are welcome and encouraged to discuss homework with other students, but you must write your solutions individually.
The course grade will be determined as follows:

- Homework (25%). Will be assigned every class;
**due every Friday**before class. We will spend a half of each Friday class discussing the solutions. One homework with the lowest score will be dropped. Scroll down to see the homework. - Midterm Exam 1 (20%), Friday, October 16, in class. Books and calculators are allowed.
- Midterm Exam 2 (20%), Friday, November 20, in class. Books and calculators are allowed.
- Final Exam (35%), Friday, December 18, 1:30 - 3:30 pm, in USB 2260. Books and calculators are allowed.

**Missing/late work:**
**Missed exams** - there will be no make-up for the exams for any reason.
A missed midterm exam counts as zero points, with the following exception.
If you miss a midterm exam due to a documented medical or family emergency,
the exam's weight will be added to the weight of the final exam.
**Late homework** will not be accepted.

**Lecture Schedule and Homework:**
It is a useful practice to read ahead the sections to be covered.

- Wednesday, September 9
- Section 1.2: Principle of Mathematical Induction (without proof); examples.
- Study preliminaries: Section 1.1 (sets) and Appendix A (logic). Study Principle of Mathematical Induction with an arbitrary base instead of 1 (Statement 1.2.3).
- Do problems 1-5; they are due Friday, September 18.
- Friday, September 11
- Section 1.3: Countable sets and its properties; countability of the set of rational numbers. Note: please correct the statement of the criterion of countability given in class: (i) should be "S is a countable set" instead of "countably infinite".
- Do problems 6-7; they are due Friday, September 18. Problems 4,5,6,7 will be graded.
- Monday, September 14
- Section 1.3: Countable union of countable sets; uncountability of the power set of N; uncountability of R by Cantor's diagonalization method.
- Home reading: Sections 2.1-2.2 (simple properties of real numbers). You do not need to memorize all those axioms and properties, but you should be very comfortable with the material in those sections. Pay attention to arithmetic-geometric mean inequality, Bernoulli inequality, and especially triangle inequality and its consequences.
- Do problems 8-9; they are due Friday, September 25.
- Wednesday, September 16
- Section 2.3: supremum and infimum. Study neighborhoods at home (Def. 2.2.7 and discussion below).
- Friday, September 18
- Section 2.4.1: properties of supremum and infimum. Study 2.4.3 (Archimedian Property).
- Do problems 10-16; they are due Friday, September 25.
- Monday, September 21
- Applications of Completeness Axiom: existence of the square root (Theorem 2.4.7); nested intervals (Theorem 2.5.2). Study density of Q (Theorem 2.4.8). Modify the argument given in class to show that there exists a real solution to the equation x^2 = a for every a > 0.
- Do problems 17-19; they are due Friday, September 25. Problems 9c, 12, 15, 17a, 19. will be graded. Problem 9c = 5 points, all others = 10 points each.
- Wednesday, September 23
- Sequences and their limits (Section 3.1).
- Do problems 20-21; they are due Friday, October 2.
- Friday, September 25
- Three remarkable limits (Examples 3.11 b, c, d).
- Do problems 22-23; they are due Friday, October 2.
- Monday, September 28
- Limit theorems (Theorems 3.2.2, 3.2.3).
- Do problems 24-26; they are due Friday, October 2. Problems 20b, 20c, 21, 22, 26 will be graded; each problem = 10 points.
- Wednesday, September 30
- Squeeze theorem (what we did in class roughly corresponds to Theorems 3.2.4, 3.2.5, 3.2.7). Examples.
- Please study Theorems 3.2.10 and 3.2.11, which we have not covered in class.
- Do problem 27; it is due Friday, October 9.
- Friday, October 2
- Monotone Convergence Theorem (Section 3.3). Application to convergence of recursively defined sequences.
- Study Sections 3.3.4, 3.3.5.
- Do problems 28-31; they are due Friday, October 9.
- Monday, October 5
- Euler's number e as the limit of the sequence (1+1/n)^n. (Section 3.3.6.) Subsequences (Section 3.4).
- Do problem 32; it is due Friday, October 9. Problems 27 a, c, d; 29; 31, 32 will be graded. Point values: 21 a, c, d = 5 points each; 29, 31, 32 = 10 points each. Total = 45 points.
- Wednesday, October 7
- Bolzano-Weierstrass Theorem (3.4.8). Limit superior and limit inferior.
- Do problems 33-34; they are due Friday, October 16.
- Friday, October 9
- Review for Midterm.
- Do problem 35; it is due Friday, October 16. Problems 33, 34, 35 will be graded; each problem = 10 points.
- Here are some practice exams. This one is most representative; the actual exam will be just a bit harder. This exam is too easy but is worth practicing; here are solutions. In this exam, try problems 1 and 7a. Additionally, it is useful to study the worked out examples in the textbook, from each section we covered. There are also problems at the end of each section; they arranged in the ascending level of difficulty. Our exam will be mid-level difficulty, so problems in the middle are most representative.
- Monday, October 12
- The Cauchy Criterion. Contractive sequences (Section 3.5).
- Wednesday, October 14
- Infinite limits (Section 3.6). Series. Example: geometric and harmonic series (Section 3.7).
- Do problems 36-41; they are due Friday, October 23.
- Friday, October 16:
**Midterm Exam 1**. Solutions. - Do problems 42-43; they are due Friday, October 23. Problems 36, 37, 39, 40, 42 will be graded. Each problem = 10 points.
- Wednesday, October 21
- Comparison tests for series (Section 3.7)
- Do problems 44-47; they are due Monday, November 2. (This is not a typo.)
- Friday, October 23
- Limit of a function (Section 4.1)
- Do problems 48-49; they are due Monday, November 2. Problems 45, 47, 48, 49 will be graded. All problems = 10 points each.
- Monday, October 26
- Limit theorems (Section 4.2)
- Wednesday, October 28
- Limit theorems continued (Section 4.2). Some classical limits.
- Friday, October 30
- Extensions of the concept of limit (Section 4.3). Continuous functions (Section 5.1).
- Do problems 50-54; they are due Friday, November 6.
- Monday, November 2
- Lipschitz functions. Combinations of continuous functions (Section 5.2). Boundedness Theorem (5.3.2).
- Do problems 55-56; they are due Friday, November 6. Problems 50 b,d, 52, 53, 56 will be graded. All problems = 10 points each, total = 40 points.
- Wednesday, November 4
- Maximum-minium Theorem (5.3.3), Intermediate Value Theorem (5.3.7), preservation of intervals (5.3.9).
- Do problems 57-59; they are due Friday, November 13.
- Friday, November 6
- Uniform Continuity (5.4.3), Continuous Inverse Theorem (5.6.5).
- Do problems 60-61; they are due Friday, November 13.
- Monday, November 9
- Differentiation: definition and basic properties of the derivative (6.1.1-6.1.5). Linearization of functions.
- Do problems 62-63; they are due Friday, November 13. Problems 57, 58, 59, 62, 63 will be graded; each = 10 points.
- Wednesday, November 11
- Landau notation (not in textbook, see e.g. this Wikipedia article). The chain rule. Derivatives of trigonometric functions.
- Do problems 64-65; they are due Friday, November 20.
- Friday, November 13
- Derivative of inverse functions (6.1.8). Local extrema (6.2.1). Rolle's Theorem (6.2.3) and Mean Value Theorem (6.2.4).
- Do problems 66-67; they are due Friday, November 20. Problems 64 (a,c), 65, 66, 67 will be graded; each = 10 points.
- Monday, November 16
- Some consequences of the Mean Value Theorem (6.2.5 - 6.2.7). L'Hospital's Rule (6.3).
- Wednesday, November 18
- Taylor's Theorem (6.4.1). Taylor series.
- Do problems 68-70; they are due Monday, November 30.
- Friday, November 20:
**Midterm Exam 2**. Solutions. - Here are some practice exams. This exam is most representative; here are solutions. In this exam, try problems 2, 3, 4, 6. Additionally, it is useful to study the worked out examples in the textbook, from each section we covered. There are also problems at the end of each section; they arranged in the ascending level of difficulty. Our exam will be mid-level difficulty, so problems in the middle are most representative.
- Monday, November 23
- Riemann Integral: definition and basic properties (Section 7.1).
- Do problems 71-73; they are due Monday, November 30.
- Wednesday, November 25
- Sequential and Darboux criteria of integrability. Class notes.
- Monday, November 30
- Integrability of continuous and monotone functions, restrictions and combinations (roughly 7.2.7 - 7.2.10). Class notes.
- Wednesday, December 2: Prof. Mark Rudelson is teaching this class.
- The plan for the rest of the course is to cover Sections 7.3, 9.1, 9.2 and 9.3.
- Do problems 74-75; they are due Wednesday, December 9.
- Friday, December 4: Prof. Jinho Baik is teaching this class.
- Returning to series. (?)
- Do problems 76-81; they are due Wednesday, December 9. Problems 76, 77, 80, 81 will be graded. Each problem = 10 points.
- Monday, December 7: Prof. Alon Nishry is teaching this class.
- Absolute and conditional convergence (9.1.1-9.1.2). Limit comparison test and Root test for absolute convergence (9.2.1-9.2.3).
- Wednesday, December 9
- Ratio Test (9.2.4-9.2.5) and Integral Test (9.2.6).
- Do problems 82-84; they are due Monday, December 14.
- Friday, December 11
- Alternating series (9.3.2). Grouping (9.1.3) and rearrangement (9.1.5) of the series
- Monday, December 14
- Review for Final Exam.
- Friday, December 18:
**Final Exam:**1:30-3:30 pm in USB 2260 - Here are some practice exams. This exam is most representative; here are solutions. (I have posted it already before the midterm exam, but Problem 1 about series was not accessible back then.) This exam was also posted before, you should be able to solve all problems there. In this exam, try all problems except 6,7,12. And in this exam, try all problems except 7,9,11. Additionally, it is useful to study the worked out examples in the textbook, from each section we covered. There are also problems at the end of each section; they arranged in the ascending level of difficulty. Our exam will be mid-level difficulty, so problems in the middle are most representative.

**Course webpage:**
http://www-personal.umich.edu/~romanv/teaching/2015-16/451/451.html