# Lecture Notes

Here are scanned notes from lectures The class discussion includes material covered on the board and in response to questions and comments that get raised during the lecture. Students are strongly encouraged to rewrite these notes carefully in their own words to make sure they understand what is here. If you have to miss a lecture, it is also a good idea to look at the notes taken in lecture by a classmate.**Lecture Notes Set 1, Wednesday, January 17**

**Lecture Notes Set 2, Friday, January 19**

- Section 1.2;
**Lecture Notes Set 3, Monday, January 22****Lecture Notes Set 4, Wednesday, January 24****Lecture Notes Set 5, Wednesday, January 31**

- Section 1.3 (Proof Technique I: working Backwards, the contrapositive, and an example thanks to Tadashi Tokieda.)

- Section 1.4 (Proof Technique II)

**On Friday, January 26 and Monday January 29**we continued through the above posted notes, finishing the topis in Lecture Notes #4 on Monday, January 29th. The lecture notes posted here contain more examples than we were able to cover in lecture and the hope is that students find it helpful to have access to the additional examples in the lecture notes posted here. I will post here the next two sets of lecture notes in case students want to read ahead (along with the reading they are doing in the textbook in advance of the lectures). We have had a ton of extraordinarily illuminating and constructive questions and comments during lecture and it would be really, really great if that continued!!!

**Lecture Notes Set 5**

**Lecture Notes Set 6**- Section 2.1 (Basic Set Operations)

**Lecture Notes Set 5, Friday, February 2, 2018**

- Section 2.1 (Basic Set Operations)

**Lecture Notes Set 6, Monday, February 4, 2018**

- Section 2.2 (up through Relations)

**Lecture Notes set 7, Wednesday, February 6th**

- Several review problems, in preparation for the midterm this Thursday on Chapter 1. (Notice please: we have posted two different practice exams on the main course homepage for you to examine - see the Announcements part of that page!). We then covered the remains of Section 2.2. (up through Equivalence Relations). Warning: the first power set on page 1 of this lecture note set is missing curly braces on right hand side!

**Lecture Notes set 8, Friday, February 8th**

- We start here talking about functions (section 2.3 of the text). There is a very abstract definition of a function (this holds that a function is a particular type of relation) and there are other ways of thinking about functios. We ask that you get comfortable with these multiple points of view and work problems that involve all these points of view. The homework includes a good variety of such problems. To make this point another way: Many people (including me) will find the discussion of functions in terms of relations as rather more abstract than they expected. That's ok - it's a great exercise to get comfortable with this.

**Lecture Notes set 9, Monday, February 11th**

- More on Functions (finished section 2.3 of the text).

**Lecture Notes set 10, Wednesday, February 13th**

- Example problems with injectivity and surjectivity; Started 2.4 of the text: Cardinality- a rigorous tool for studying infinite sets. For a far less rigorous but still catchy treatment of the infinite and the heroic role played by 0 therin,see:

you can count as high as you could ever go.

Forever.

Towards infinity.

No one ever gets there but you can try"

**Lecture Notes Set 11, Friday, February 15th**

- More on cardinality - some additional tools for proving countability. (section 2.4 of the text). .

**Lecture Notes set 12, Monday, February 18th**

- Finished up cardinality. Outlined section 2.5 (Axioms of Set Theory) which will not be on any exam or quiz or homework (but which might be covered carefully at the end of the term, time allowing). We discussed our course's aim to introduce tools for rigor, and the role rigor plays both in Math and other Sciences to advance
knowledge. (For a nice example, see the summary of the role rigorous mathematics played in a fundamental breakthrough in Science announced in 2016 found in this article. Note: this breakthrough was especially newsworthy this past fall.) We also discussed what it means for a statement (e.g. the continuum hypothesis) to be independent of a field's axioms (e.g. the Zermelo-Frankel axioms of set theory). We emphasized the analogy with geometry - where Euclid's parallel postulate is known to be independent with the other axioms on which Euclidean geometry is based. We talked about how the negation of the parallel postulate is absolutely appropriate in many settings (e.g. in spherical geometry, or in extraordinarily relevant theories of gravity). For a nice discussion of the parallel postulate see here. Question: How would you know if the
world where you are working is one where the parallel postulate is in play or not? Answer: we spoke about how (surprisingly!) the Pythagorean Theorem can be a
good canary in the coal mine..

**Lecture Notes 13, Wednesday, February 21rst**

- Section 3.1: Mathematical Induction.

**Lecture Notes 14, Friday, February 23rd**

- More examples of
Mathematical Induction, start of Section 3.2 (Ordered Fields).

**Lecture Notes 15, Monday, February 26th**

- More on ordered fields.

**Lecture Notes 16, Monday February 26thth**

- Finished discussion and examples on ordered fields, started Section 3.3 (related to the completeness of the real numbers). We discussed an example of a strategy that is often used for proving indequalities called 'give yourself an epsilon of room' - a phrase made popular by Professor Terence Tao (UCLA) (for lots of very good examples of where this strategy comes into play outside the immediate scope of this class see the chapter starting on page 323 of this text by T. Tao, it's also here.
We mentioned when reviewing what a field is that there are two operations - addition and multiplication - and they are really distict. It's important to know that multiplication of integers can be thought of as repeated addition, but it's also important to know that multiplication of real numbers is a distinct operation from addition. (In the rigorous, complete construction of the Field of Real numbers, multiplication of reals is built from multiplication of integers but involves other - distinct - ideas as well.) See 10:35 - 14:50 of this presentation by Professor Paul Sally (UChicago) for an example of a simple geometric situation where multiplication of irrationals arises immediately.
Regarding the rigorous construction of the real numbers: if we had more time, we would start with the Natural numbers $N$ where we would assume just a few axioms (called the Peano Axioms) to establish the basic properties of the natural numbers. From there, we would construct the integers $Z$ and establish all the familiar properties for arithmetic there (we would prove them as Theorems, using the Axioms for $N$), and then we would build from this the rational numbers $Q$, proving Theorems that establish the basic properties of addition and multiplication in Q(adopting no more axioms beyond those we already are holding from $N$). The final step we would take would be to construct the real numbers $R$, and prove all the properties that the book calls `Axioms". As mentioned, we simply
don't have time to do this - and choose instead to follow the path set forward by our very good textbook! Those students in 3283w wishing to see this careful construction of the reals from just a few very basic axioms for N can look in many places. Two references would be this textbook by Professor Steven Krantz (Washington University) or these notes by Professor Joseph Taylor (Utah).
**Lecture Notes 17, Wednesday February 28th****Lecture Notes 18, Friday, March 2****Lecture Notes 19, Monday, March 5****Lecture Notes 20, Wednesday, March 7****Lecture notes 21, which we will cover in the next lecture that we have (after spring break). Lecture notes 21 covers the Heine-Borel theorem (another topic having to do with Compact sets.) Note that the huge majority of Lecture Notes 21 is devoted to the proof of the Heine-Borel theorem. On first reading, it's really important that you remember the statement of this Theorem (and some of it's consequences - especially the ones presented in the last part of lecture Notes 21.)****Lecture Notes 21, Monday, March 19****Lecture Notes 23, Wednesday, March 20****Lecture Notes 23, Friday, March 22****Lecture Notes 24, Monday, March 26****Lecture Notes 25, Wednesday, March 28****Lecture Notes 26, and****Lecture Notes 27,Friday, March 30****Lecture Notes 28,Monday April 2**
Finished section 4.3, started section 4.4 on Subsequences. **Lecture Notes 29,Wednesday April 4**
Finished section 4.4 on Subsequences. You can find more worked examples on this material in the first few sections of **Lecture Notes 31,Friday April 6. We finished this 31rst set of lecture notes after doing a warm-up discussion on Cardinality and on the difference between a necessary condition and a sufficient condition for something to be true. (For example: for a series to converge, it is a necessary condition that the terms go to zero. This is definitely NOT a sufficient condition because we gave an example (the harmonic series) of a series whose terms go to zero but which does not converge. It is important that students keep this in mind. (Every term, several students will forget this distinction on the final midterm and the final exam. We are really hoping to make the point clearly enough and often enough this term so that this confusion is eliminated.)****Lecture Notes 32**for a preview of the material we will cover on Monday. d**Lecture Notes 32Monday April 9. Section 8.2 (Convergence tests)****Lecture Notes 33,Wednesday April 11. More on Section 8.2 (Convergence tests, Absolute convergence, alternating series.)****Lecture Notes 34,Friday April 13. Section 8.3 (Power Series and Start of Section 5.1 - discussion of limits of functions).****Lecture Notes 35 (Consists entirely of practice problems with sequences and other past material - most of this was not discussed in class, provided here for your furthur practice.),Monday, April 16.. We spent most of the time (all but about 7 minutes, actually) reviewing for the midterm. The way we did this was to discuss in detail past exams. We started by looking at problems 1a and 5 from the second practice midterm that we posted for Midterm 2 at our site here for this class. Then we looked at a second trove of practice problems which was Midterm 3 from the Fall 2017 instance of 3283w. That second pool of problems is now posted here too, along with solutions, as a second practice midterm for Midterm 3. (It goes without saying, but we say it anyways, that we have already posted a first practice midterm for midterm 3, along with solutions.) In the last 7 minutes of class we discussed two features of the definition of a limit of a function at a point c which I think don't always get noticed the first time a person reads that definition. First of all: we will only define the limit at accumulation points of the domain (we are following the text's convention here, and it's reasonable!). Second, the value of the funcition at the point c where you are taking the limit has no influence whatsoever on the value of the limit - in fact, the function doesn't even have to be defined at that point c. (See end of Lecture notes 34 for more on this.)****Lecture Notes 36. Wednesday, April 18. The notes here consist of more practice problems for tomorrow's midterm, and also a discussion of section 5.2 of the text (Limit Laws.). As announced in class, by email, and also posted on the main page for this course, the midterm covers sections 3.4, 3.5, 4.1, 4.2, 4.3, 4.4, 8.1, 8.2. We have focussed on the most recent material in this review because we have had the opportunity to review other material earlier. Here is a set of notes that review some of the material from section 3.4:****Section 3.4 review - some notes on topology of the reals.****Lecture Notes 37. Friday, April 20. More properties of continuous functions and start of Section 5.3.****Lecture Notes 38. Monday, April 23. Finished Section 5.3 (proof of Intermediate Value Theorem and Proof that the continuous image of a compact set is compact. The latter result implies that on a compact set any continuous function attains its maximum and it attains its minimum. Started section 6.1 (differentiation). The material we did on differentiation is in****Lecture Notes 39. Please note that section 5.4 won't be included in the course. That topic - Uniform Continuity - is much better reserved for the next course in Analysis that you take - e.g. Math 5615H. This topic is not used materially in the rest of our text and certainly has no impact on what we aim to study in our last two weeks here in 3283w.****Lecture Notes 40. Wednesday, April 25. We finished the material in Lecture Notes 39 and started the discussion of the Mean Value Theorem. .****Lecture Notes 41. Friday, April 27. More on the Mean Value Theorem including applications. L'Hospitals Theorem.****Lecture Notes 42. Monday April 30. First worked to finish lecture notes 41. Then moved into Lecture Notes 42 on Taylor's Theorem. (Ignore please the note here saying that Taylor's Theorem will not be on the Final: it could be!!) One way to think of this result is as a generalization of the Mean Value Theorem! One way to think about proving it is ``integration by parts". Another way to think about proving it is described nicely on Professor Timothy Gower's website (at Cambridge) here..****Lecture Notes 43. Wednesday May 2. Finished discussion of Taylor's Theorem and used Taylor Polynomials to motivate why L'Hospitals rule might be true (or at least, why it has a better chance of being true than some other rules we might try and sell to the unsuspecting). We walked through an argument that showed the relationship between Taylor's Theorem and Integration by parts. Worked a number of examples of problems similar to homework problems and notes for that discussion are included here as "Lecture Notes 43.". `**

- The new material here connected to the Completeness of the real numbers begins with the notions of lower bounds, upper bounds, greatest lower bounds ("inf") and least upper bounds ("sup"). We mentioned how the notion of "sup" is at least a little bit like the description of Jack Sparrow as `without a doubt the worst pirate I've ever heard of'. (The 'sup' of a set is the lowest possible upper bound.....but it's still an upper bound!)

- Section 3.3: the Completeness of the real numbers. We emphasized that the real numbers enjoy a wonderful property called "Completeness". It is no exhaggeration to say that this property is the basis for an incredibly large part of the development of mathematical techniques and is directly responsible for many of the useful applications of mathematics to the physical, biological, and social sciences. (For example, most every time we talk about the existence of a solution to a difficult to solve ODE that models some application, we are relying on the completeness of the real numbers.) We are super fortunate that the real numbers exhibit this property! In class we talked about why it's possible to argue that it's strange - truly strange - to label this property an "Axiom" on the first pass through this material. It is a little bit like talking about your favorite restaurant on Snelling Avenue Northeast of the U of M with a friend. If that friend asks you, "Does it have take-out options?" and you answer, "Let's assume it does,it's a restaurant!" you would be forgiven for your direct speech. If your friend asked "Can you get water with your meal there?" and you answered "Let's assume you can, it's a restaurant!" you would again be forgiven for your direct way of talking. But if your friend asked you "Does it serve tasty Mapo Tofu?" and you answered "Let's assume it does, it's a restaurant!" you would come across as sounding truly confused and unhinged because that is not something you usually ASSUME when you talk about your favorite restaurant - and it's certainly not something you present as an assumption the first time you talk about the wonders of this restaurant with your friend. Rather, the presence of this wonderful property ("It has tasty Mapo Tofu!") is something you
celebrate as being one of the truly great features of the restaurant.

How to explain then the presence of the term "The completeness Axiom" in our book? Our textbook's use of the phrase ``The completeness axiom" is connected with the following fact, which is not (to my knowledge) presented (let alone proven) in the text: There is really only one complete, ordered field (and the real numbers is it!!). To put this slightly more precisely, we'd say that up to a relabelling that respects the order relation and the properties of addition and subtraction, the real numbers are the only set that is an ordered field which enjoys the completeness property. So, it makes sense to talk about "completeness" as an Axiom if you are interested in learning if there is more than one field out there like the reals. (You would assume that someone else has constructed a set and assume that set is an ordered field and assume that it satisfies the completeness property. Then you would show that this supposedly exotic new set is really nothing but the real numbers (with possibly different labels on the set elements).) Such a discussion is not for this course (and as far as I can see, not in the textbook). To continue the restaurant metaphor: if you wanted to prove that the business named "Szechuan" in Roseville is the unique restaurant on Snelling avenue which serves tasty Mapo tofu, then it would be appropriate for you to say something like, "Suppose you have in mind a hypothetical restaurant on Snelling avenue and assume this hypothetical restaurant has good Mapo Tofu." and then follow up that set-up by showing that the hypothetical restaurant must be this same restaurant named Szechuan. Such a discussion is not necessary (and in some sense not even reasonable) if we find ourselves on Snelling avenue for the first time and simply want to go get some Mapo Tofu.

Those students who would like to know precisely what it means to say that the real numbers are the unique complete ordered field can see this book (starting on page 91 of Volume 1) by Professor Garling (Cambridge University), or these notes by Professor Gamelin (UCLA) which give a sketch of the result starting on page 9 (these notes also give a nice survey of some different rigorous models of the real numbers that have been constructed.). .

- Section 3.4 (Topology of the Reals)

- Section 3.4 (Finished discussing this section on Topology of the Reals), also advertised these practice problems that might help you start your homework problems on this material. You will find there a few parts of problem 2 from the section worked in detail.

- Section 3.5: Started talking about compact sets in the reals. Worked these problems about closures and accumulation points as a warm up.

- We got to the statement of the Heine-Borel Theorem today.

- Finished Section 3.5 and started in on Section 4.1 (Sequences: definition of convergence , examples.) If we have time, we will start in on Section 4.2 and Lecture Notes 24
In discussing the second midterm (and discussing how best to plug into the course) we discussed double stars in the big dipper (Mizar and Alcor). and the biological fact that these are most easily seen by looking a little bit away from them.

- Discussed Section 4.1 (Sequences: definition of convergence , examples.)

- Finished Section 4.1 and started Section 4.2 (Limit Laws.)

- Finished Section 4.2. Changed homework due tomorrow (problems from 4.3 are postponed to next week.)

- Discussed infinite limits, also Section 4.3. Examples here of working with Monotone sequences and the start of a discussion of Cauchy sequences.

**Lecture Notes 29 (these notes are also about Section 4.4.)**

**Lecture Notes 30**. Be careful about Lecture Notes 30 though - they end with some material on Section 5.1 and instead our course will move next to discuss Series. (This is not the order in the textbook, but it is a very common order of things in other textbooks and we will follow it here, as we described in the course schedule distributed at the start of the term. See

**Lecture Notes 31**for a preview of the material we will cover on Friday.