Office hours: Tuesdays and Thursdays 5:00PM - 6:00PM and 8:30 - 9PM, Hill 624 or by appointment.
Email: cl.volkov at rutgers dot edu (for friends) / fq15 at scarletmail dot rutgers dot edu (for teaching)
In the Summer of 2017 I taught 640:311 (Introduction to Real Analysis I) for Sections T6.
Lecture 1 (May 30, 2017). Lecture Notes
For more details, please read Ted Sundstrom, Mathematical Reasoning: Writing and Proof, Version 2.1, Chapter 1 and 2.
Also you can read Zorich, Mathematical Analysis I, Section 1.1.
Lecture 2 (June 1, 2017). Lecture Notes, Workshop 1 (written by Dr. Scheffer), Writing Samples.
The course materials mainly comes from Chapter 5 and 6 of Sundstrom's book.
Also you can read Zorich's book, Section 1.2 and 1.3.
All workshops are due 11:55PM the next Tuesday. So in case you have questions, you can discuss with me either before or after Tuesday's class.
Lecture 3 (June 6, 2017). Lecture Notes
For more details, please read Zorich, Section 2.1.
An slightly different argument showing root 2 is not rational can be found in [Z], 2.2.2.c. The argument in the notes is modified from [A], Theorem 1.4.5.
The construction of real numbers using Dedekind cuts can be found in [A], Section 8.6.
Lecture 4 (June 8, 2017). Lecture Notes, Workshop 2
Since I wasn't able to cover the density theorem, the workshop problem 5 is removed from this week's assignment.
By now you should finish reading [A], Section 1.1 - 1.3 and Thompson-Brucker-Brucker, Elementary Real Analysis, Section 1.1 - 1.7.
Lecture 5 (June 13, 2017). Lecture Notes
It is very important that the Nested Interval Property applies only to closed intervals that are bounded. Think: which part of the proof fails when the intervals are not bounded.
One can prove under the assumption of Archimedean Property, Nested Interval Property can imply Axiom of Completeness. Please see James Propp's paper Real Analysis in Reverse for more details. In the coming Chapter we will see a lot more such properties.
Lecture 6 (June 15, 2017). Lecture Notes, Workshop 3
In case you are interested in solving the optional workshop problem, please see the Notes on Countable Sets and Cantor's Diagonalization.
The idea of Cantor's Diagonalization is to construct a decimal that is outside of the range of the function from the naturals to reals. Please see [A], Section 1.6 for details. In the note above you will find the most essential argument.
By now you should finish reading Section 1.4 - 1.5 and 2.1 of the textbook, and Section 1.8 - 1.10, 2.1 - 2.4 of the TBB book
About cardinalities, please read [Gamow] One Two Three Infinity, Chapter 1 and 2.
Lecture 7 (June 20, 2017). Lecture Notes
Here you should learn the technique of finding the N from the given conditions of convergence, instead of from the estimates.
Also, to use the Algebraic Limit Theorem, it is important to make sure that all the limits involved exist. Otherwise you might make some serious mistakes.
Lecture 8 (June 22, 2017). Lecture Notes, Workshop 4
For the Order Limit Theorem, it is important to make sure that all the limits involved exist. Otherwise you might make some serious mistakes.
Monotone Convergence gives a very convenient way of proving convergence, but usually does not tell you directly what the limit is. In general, getting the actual limit is usually difficult. In this class we only deal with some simple cases.
Please make sure you can recall how to prove AoC implies MCT. Make a brief summary definitely helps.
By now you should finish reading [A] 2.2 - 2.4, [TBB] 2.5 - 2.10.
Lecture 9 (June 27, 2017). Lecture Notes
In case you are struggling with the Workshop 4, Mr. Yang kindly wrote a guide to all the problems and agreed to share. Note that this is just a guide. The thinking process has been elaborated presented. Yet it does not make a proof. You still need to organize these thoughts into a proof.
Lecture 10 (June 29, 2017). Lecture Notes, Workshop 5
In case you are not satisfied with certain grade of the quizzes, or you have missed it due to any reason, please finish a write-up of the homework of the previous lecture and present your solution to me in person.
For example, if you are not happy with your grades for Quiz 7, then you should do all the homework problems assigned in Lecture 7.
I'll check a random problem to see if you really have good understanding towards it. If you have, then your quiz grade will be made to 8/10. To make up quizzes 1 - 9, your solutions must be presented before July 13th. After July 13th, the grades for Quiz 1 - 9 cannot be changed any more.
By now you should finish reading [A] 2.5 - 2.6, [TBB] 2.11 - 2.12.
Lecture 11 (July 4, 2017) No lectures today. Happy holiday!
Lecture 12 (July 6, 2017). Midterm Exam, Workshop 6 (Written by Dr. Scheffer)
Second chance policies: In case you didn't do well in the midterm, here is what you should do:
Lecture 13 (July 11, 2017). Course Notes
For those who missed tonight's lecture, please make sure you are capable of proving every single entry in the table on Page 9. In class I explained those examples on the blackboard. However the proof was only given orally. Please let me know if you are having trouble proving any items. I will be happy to supply an argument.
The written quiz tonight is replaced as a Questionnaire regarding the midterm. Please find it in Sakai Assignments.
Lecture 14 (July 13, 2017). Course Notes, Workshop 7
Note: You don't need to worry the compactness part in either [A] or [TBB]. I did use the examples in [A] and the motivating comments in [TBB]. For Workshop 7, you don't need to know anything other than the currently posted course notes.
By now you should finish reading [A] 3.2, [TBB] 4.1 - 4.4.
Lecture 15 (July 18, 2017). Course Notes
I have set up the system, so Workshop 6 can be (re)submitted until Aug. 4. Workshop 7 can be (re)submitted until July 25th.
Lecture 16 (July 20, 2017). Course Notes, Workshop 8
By now you should finish reading [A] 3.3, [TBB] 4.5 (Note that the Cousin's Property was not covered). You should start reading [A] 4.2 and [TBB] 5.1.
Sorry for having delivered a stupidly organized lecture tonight. Hopefully the reorganized notes look better. Please let me know if you have troubles.
Lecture 17 (July 25, 2017). Course Notes
Here are the pictures of the blackboard: Sequential Criterion - What we want, Sequential Criterion - What we know
Lecture 18 (July 27, 2017). Course Notes, Workshop 9
By now you should finish reading [A] 4.1 - 4.3, [TBB] 5.1, 5.2, 5.4 and 5.5.
On the second page of Workshop 9 you will find some comments to the exercises in [A]. Please at least attempt those problems I boldfaced.
Lecture 19 (Aug. 1, 2017). Course Notes
As we are about to finish Chapter 4 on Thursday, it is a very good point to review everything. If you have a good understanding on the materials in Chapter 1 to 4, you should feel no difficulty at all to understand Chapter 5, and most of the parts in Chapter 6 (until you arrive at the issue of uniform convergence of sequences and series of functions). If you are taking 312 next semester, your life will be easy for a while. So please do so without hesitation.
For those who didn't do well in the quiz tonight, please answer the following questions:
1. How many exercises did you attempt in 3.2, 3.3, 4.2, 4.3, 4.4?
2. What kind of difficulty did you experience?
3. Anything I can do to help?
Please send your answers through emails. The grade for the quiz will be adjusted to 8/10 or your actual grade, whichever is higher.
Lecture 20 (Aug. 3, 2017). Course Notes, Workshop 10
Please attempt to prove those facts in Part 3 by yourself and do not read my argument unless you have no clue. My argument might be too complicated than it should be. The easiest way to simplify any complicated argument is to work your own argument without reading a word from the original one.
The reason I chose these two easy problems for this last workshop assignment is to provide more free time for you to review the materials and attempt all the other problems in the book. Don't be lazy. You are not studying analysis for me, but to prepare for your future studies. The exercises in [A] is really the minimal amount you have to go through in order to master the skills.
By now you should finish reading [A] 4.4 - 4.5 and [TBB] 5.6 - 5.9.
Lecture 21 (Aug. 8, 2017). Course Notes.
As you can see, if you have a solid understand for Chapter 1 through 4, there is no trouble for you to understand at least the theory of derivatives. The main challenge for this Chapter is how to use the results in real life. Please see Zorich's exercises for more practice.
Lecture 22 (Aug. 10, 2017). Course Notes. Review of Chapter 1 to 4
The exam will be held on next Tuesday. There will be 13 problems with 200 points. 150 points are considered as a perfect score. Please find more details on Sakai.
By now you should finish reading [A] 5.1 - 5.3. If you have time, please also read [TBB] 7.1 - 7.7. We don't have enough time covering all these materials however the knowledge will be assumed in 312.