Real Analysis

2/5: The exercises assigned today are due on 2/10 except as noted. Written exercises should be written neatly or typed (not in your notes) and turned in at the beginning of class on the indicated date. Of course, there is nothing to turn in for reading assignments, but there could be a quiz or writing assignment over that material. Read the notes for insights into style and word choice. Exercises with a (T) to the left of the number may be written in two-column format as described in class. Exercise without a (T) need to be written in paragraph format as demonstrated in the notes.

You are welcomed and encouraged to stop by my office for help on the homework. However, you may not ask questions over homework on the day that the assignment is due.

Do NOT assume that any assignment beyond today's is correct! I often change them the day of class.

* DO THIS BEFORE OUR NEXT CLASS: Read the Vital Information section of the notes, pages 1-14. Browse the Index too so you get an idea of what's there.
* DO THIS BEFORE OUR NEXT CLASS: A comment on analysis homework - please read this carefully.
* Read the Academic Honesty Guidelines. There is a link in the header above.
* DO THIS BEFORE OUR NEXT CLASS: Read section 1.1 in the notes.
* DO THIS BEFORE OUR NEXT CLASS: Do exercises 1bdef from Section 1.1. Do not use two-column format on this exercise. Remember that your proofs need to include words and justifications, not just computations!
* DO THIS BEFORE OUR NEXT CLASS: Check the blog for new entries. Even if I fail to put this in your assignment, it would be a good idea to check the blog at least every other day. As I read your homework or think about how class went, I will post insights and hints in the blog.
* If you wish, you can download and install Mathematica on your personal computer, and have access to Mathematica Online. Links to the downloads are on this page.

Due 2/10, written very neatly, in complete sentences. Read the notes for insights into style and word choice.
* Check the blog for new entries.
* From section 1.1 in the notes, do exercises 3, 4bef, 6. You may use Exercises 1bdef for these proofs!
* Read ALL of the exercises in section 1.1! In section 1.2 and beyond, you may need those results and you can use them!
* Optional: Sign up for your free MAA membership at this link.

2/10: Due 2/14. Today's Mathematica File

* Read section 1.3 in the notes.
* There are interactive induction proofs on this page. Work them through, filling in each step and checking your work. You do not have to turn these in; it's part of your reading assignment.
* Read all of the exercises in section 1.3, especially 5, 6, 7, 8, and 14-19. We will use some of those inequalities.
* Do exercises 1bd, 2g, 5bd, 18 from section 1.3 in the notes. Model your proofs after Examples 1.3.14, 1.3.21, and 1.3.29, in terms of language, amount of detail, and format.
* Check the blog every day for new entries.

2/12: Due 2/17. Last day to add or drop a course.

* Read section 1.4 in the notes.
* Work through this proof of one part of DeMorgan's Laws. There is nothing to turn in here - this is part of your reading assignment.
* Read all of the exercises in section 1.4, especially exercises 21 and 22. We will use these in a future section, and you may need to use them in your homework.
* From section 1.4, do exercises 1, 10, 11, 15, 16. For problem 11, skip the "under what circumstances..." part of the exercise. There is a typo.
* Check the blog every day for new entries.

* I found some minor problems with the index in our text. A few numbers are slightly off. You may want to print another copy of the index,

* Listen to this radio program. You can read the article, or click on the blue "Listen" button on the left side of the page. You can read the text as well, but listening might be more effective. Here is a link to a related video produced by Arizona State University.

2/14: Due 2/19.

* Read section 1.5 in the notes.
* Read all of the exercises in section 1.5.
* From section 1.5, do exercises 1cefg, 4, 7.
* Check the blog every day for new entries.

A friendly reminder: Although it's early in the semester, you should all know that your proofs must contain complete sentences, not just phrases and random thoughts. No scratch work should be on the page. Sentences start with words (and capital letters), not mathematical symbols, variables, or numbers. Sentences also end with periods, question marks, or exclamation points. It's not appropriate to use mathematical symbols as verbs (although an occasional \(\Rightarrow\) in a sequence of computations is OK), and abbreviations really aren't appropriate either. I expect good, polished writing. Yes, this may mean re-copying your work. Use pencil so you can erase; scribbling out words you don't like is not appropriate. Missing punctuation really does matter!

2/21: Due 2/26.

* Read section 1.8 in the notes.
* Read all of the exercises in section 1.8, especially 2, 10, 11, 17.
* Do exercises 1bde, 3, 7, 14 from section 1.8.
* I'm going to stop putting "Check the blog every day for new entries." in your list of assignments. You should still check the blog every day for new entries, but I shouldn't have to remind you by now.

A somewhat-less-friendly reminder: You should all know that your proofs must contain complete sentences, not just phrases and random thoughts. No scratch work should be on the page. Sentences start with words (and capital letters), not mathematical symbols, variables, or numbers. Sentences also end with periods, question marks, or exclamation points. It's not appropriate to use mathematical symbols as verbs (although an occasional \(\Rightarrow\) in a sequence of computations is OK, as long as you could replace the symbol with the phrase "which implies" and the sentence makes sense), and abbreviations really aren't appropriate either. I expect good, polished writing. Yes, this may mean re-copying your work. Also, you really don't want to assume what you're trying to prove! That can result in an immediate 1 out of 5. If you have any missing homework assignments or numerous blank problems on assignments you did turn in , I recommend re-reading the bold statement on page xi of the notes.

3/6: Due 3/25.

* Watch this introductory video.

* Read this page on homework expectations.

* Log in to the blog. Scroll down to "Log In" (on the right side) and use the information I provided in an email.

* Read section 2.3 in the notes.
* Read all of the exercises in section 2.3. You might even write out a list of exercises that seem like they could come in handy in the future; that list might include 3-8, 12, 14, 17, 22-27. Pay particular attention to 7-8, which examine what can happen when we add or multiply divergent sequences. We will need 14 and 26 in future computations as well.
* Work through this interactive example to help with this assignment.
* Do exercises 1ef, 2de, 7, 8, 11 from section 2.3. Make sure that you evaluate every limit carefully; for example, if you use Theorem 2.3.25, you need to show every step in evaluating the limit.

* Read the sheet attached to your exam and included here carefully. Heed the advice! Also, make sure you are checking the blog often.

3/23: ONLINE INSTRUCTION - FINALIZED

Due 3/27 before noon. Send me your typed assignment as an email attachment.

You can download the Mathematica code for generating recursively defined sequences here.

* In case you missed it, read this page on homework expectations. You have an assignment due Wednesday, March 25.

Required videos

* With your class text available in some form, open to Section 2.4 and watch these videos and take notes.
(1) Introduction video, definition of monotone
(2) The Monotone Convergence Theorem
(3) Example 2.4.8
(4) Example 2.4.11
(5) Example 2.4.15

Optional material

Reading Assignment

* Re-read ALL of the theorems, examples, and exercises in Section 2.3! Those can make future exercises much easier.
* Read ALL of Section 2.4 in the notes. Model your proofs after the examples, but do not include your scratch work even though I have done that in a few examples.
* Read and work this interactive example.
* Read exercise 12. (The rest are more computational.)

Homework

* Take this short quiz over the videos. This will count as an in-class writing. After you take the quiz, the answers should appear on the screen. You should also receive an email confirmation so you can check your answers.

* TYPE SOLUTIONS TO exercises 1cd, 3, 4, 13ac from Section 2.4. ONE ADDITIONAL REQUEST: Can you include your last name(s) and the section number in the file name of the document you send me? I wound up with lots of variations on "Section 2.3 Homework" and it was hard for me to tell when I had downloaded them all. Much appreciated.

3/25: ONLINE INSTRUCTION - FINALIZED

Due 3/30.

Required videos

* With your class text available in some form, open to Section 2.5 and watch these videos.
(1) The definition of a subsequence
(2) Subsequences explained
(3) The major theorems involving subsequences
(4) The limit superior
(5) Another way to think about the BW Theorem and the limit superior

Optional material

* There are many videos on this material on YouTube, but I don't recommend any one specifically. If you find one you like, let me know!

Reading assignment

* Read Section 2.5 in the notes.
* Read all of the exercises in Section 2.5, especially numbers 3, 4, and 9-16.

Homework

* TYPE YOUR SOLUTIONS TO exercises 1cd, 2ad, 3, 6 from Section 2.5.

3/27: ONLINE INSTRUCTION - FINALIZED

Due 4/1.

Required videos

* With your class text available in some form, open to Section 2.6 and watch these videos.
(1) The need for Cauchy sequences
(2) The definition of a Cauchy sequence
(3) Two important theorems
(4) The Cauchy Criterion
(5) Contrative sequences
(6) An important example

Optional material

* There are many videos on this material on YouTube, but I don't recommend any one specifically. If you find one you like, let me know!

Reading assignment

* After viewing enough of the videos that you understand the material, re-read Section 2.6 in the notes.
* Read exercises 4-8 from Section 2.6. Also, read Example 2.6.16 carefully. That relates to Exercise 11 in Section 2.3. (That's a typo in the notes.)

Homework

* TYPE YOUR SOLUTIONS TO exercises1ab, 3, 9 from Section 2.6.

3/30: ONLINE INSTRUCTION - FINALIZED

Due 4/3.

Required videos

* With your class text available in some form, open to Section 2.7 and watch these videos.
(1) The Definition of Properly Divergent
(2) Examples of Properly Divergent Sequences
(3) More Useful Examples
(4) Comparison Tests for Proper Divergence
(5) An Example Involving the Limit Comparison Test

Optional material

Reading assignment

* After viewing enough of the videos that you understand the material, Read Section 2.7 in the notes.
* Read all exercises from Section 2.7 except for 8 and 19. There re many useful results in there, and I just can't assign them all!

Homework

* TYPE YOUR SOLUTIONS TO exercises 1, 2, 5, 18ae.

4/1: ONLINE INSTRUCTION - FINALIZED

Due 4/6.

Required videos

* With your class text available in some form, open to Section 2.8 and watch these videos.
(1) An Intuitive Introduction to Series
(2) The Formal Definition of a Series
(3) Two Very Important Series
(4) Telescoping Series
(5) The nth Term Conditions
(6) Convergence Theorems
(7) A Very Powerful Test

Optional material

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 2.8 in the notes.
* Read exercise 2 from Section 2.8.

Homework

* Take this short quiz over the videos and the reading. This will count as an in-class writing. After you take the quiz, the answers should appear on the screen. You should also receive an email confirmation so you can check your answers.

* TYPE YOUR SOLUTIONS TO exercises 1, 2gh, 4, 5ad, 12a.

4/3: ONLINE INSTRUCTION - FINALIZED

The proof template

Due 4/8.

Required videos

* With your class text available in some form, open to Section 3.1 and watch these videos. Making these videos is starting to get old - sorry for the silliness in these videos. I hope it's not too annoying.
(1) Introduction to Limits of Functions at a Point
(2) The Definition of a Limit (This is posted on my Google Drive, not on YouTube. Had issues uploading to YouTube.)
(3) Important Initial Theorems About Limits
(4) A Template Proof Example
(5) Bonus! An Example That Is Not In the Notes
(6) Very Useful Techniques
(7) How to Show that a Limit Doesn't Exist

Optional material

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 3.1 in the notes.
* Read exercises 6-9 in Section 3.1.

Homework

* TYPE YOUR SOLUTIONS TO exercises 1ad, 2ah, 5d, 8, 12cd.

4/6: ONLINE INSTRUCTION - FINALIZED

Don't forget the interactive web pages - they can be very useful tools! The geometric proof of Example 3.2.13(c). Graphs of functions relevant to the material covered in today's class.

Here is a sneak peek at the first page of Exam 2!

Due 4/15.

Required videos

* With your class text available in some form, open to Section 3.2 and watch these videos.
(1) Bounded vs. Bounded at a Point
(2) Combinations of Functions
(3) The Squeeze Theorem
(4) Examples Involving Trig Functions
(5) A Very Important Example

Optional material

* Nothing really caught my eye on this topic.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 3.2 in the notes.
* Re-read the examples from Sections 2.2-2.3 to remind yourselves how to apply a "Co_" theorem, like CoS or CoF.
* Read exercises 6, 7, and 8 from Section 3.2 in the notes.

Homework

* TYPE YOUR SOLUTIONS TO exercises 1ab, 2c, 3b, 7.

4/8: ONLINE INSTRUCTION - FINALIZED

Here is a sneak peek at the first page of Exam 2! My kids said I should also share this with you to help you on the exam.

Due 4/17.

Required videos

* With your class text available in some form, open to Section 3.3 and watch these videos.
(1) One-Sided Limits
(2) Sequential Criterion
(3) As x tends to Infinity
(4) Examples
(5) Divergence

Optional material

* Not much available on one-sided limits using the definition. Most videos use an intuitive approach.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 3.3 in the notes.
* Read exercises 3-6 from Section 3.3.

Homework

* TYPE YOUR SOLUTIONS TO exercises 1ac, 2ce, 5.

4/15: ONLINE INSTRUCTION. FINALIZED

Prepare for Exam 2, a take-home style exam that will be distributed on Friday, April 17.

Here is a sneak peek at the first page of Exam 2! My kids said I should also share this with you to help you on the exam.

Due 4/24.

Required videos

* With your class text available in some form, open to Section 4.1 and watch these videos.
(1) Introduction to Continuity
(2) Continuity and Limits
(3) Template Proofs
(4) Discontinuity
(5) Dirichlet's Function
(6) The Thomae Function

Optional material

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 4.1 in the notes.
* Read exercises 11-19 from Section 4.1 in the notes. Exercises 12 and 13 are particularly useful. Check out Exercise 15 as well - you've done that one already!

Homework

* TYPE YOUR SOLUTIONS TO exercises 1abd, 2, 5, 13.

4/17: Exam 2 distributed. The exam will cover the material we learned from 2/24 - 4/8, which amounts to Chapters 2 and 3. I have included Section 2.1 on this exam, even though it was on the first exam as well

Here is a sneak peek at the first page of Exam 2! My kids said I should also share this with you to help you on the exam. Not really sure why.

This will be a take-home, well, you are at home. OK, this will be an open text exam. I will distribute the exam to you via the Help Page. Expect it to be available by 8:00 am. The exam will be due via email attachment by noon on April 22. You will work alone on this exam, and submit your work to me via email in a word-processed document. Word and pdf formats are acceptable. Specific guidelines on what resources you can use will be on the exam sheet.

4/22: ONLINE INSTRUCTION - FINALIZED. EXAM 2 DUE TODAY BY NOON.

Don't forget the interactive web pages - there is one just for studying for the third exam!

Due 4/27.

Required videos

* With your class text available in some form, open to Section 4.2 and watch these videos.
(1) CoCF - Combinations of Continuous Functions
(2) Why We Need To Improve Continuity
(3) Closed Bounded Sets Are Awesome
(4) Extreme Values
(5) The Intermediate Value Theorem

Optional material

* I didn't find any particularly good videos on this topic. If you find some, let me know!

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 4.2 in the notes.
* Read exercises 2-4, 7, 8, 13, 14, 15, 20, 21, 22 from Section 4.2. Exercise 21 is particularly important!

Homework

* TYPE YOUR SOLUTIONS TO exercises 1, 6, 20 from Section 4.2.

4/24: ONLINE INSTRUCTION - FINALIZED

Due 4/29.

Required videos

* With your class text available in some form, open to Section 4.3 and watch these videos.
(1) Uniform Continuity Introduced
(2) Continuity vs. UNIFORM Continuity
(3) Showing a Function is Not Uniformly Continuous
(4) Conditions That Imply Uniform Continuity
(5) Uniform Continuity and Cauchy Sequences
(6) Piecewise Continuous Functions
(7) Piecewise Linear Functions

Optional material

* Here is an example showing uniform continuity, and here is one showing a function is not uniformily continuous.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 4.3 in the notes.
* Read exercises 8, 9, 11, 12, 17, and 21 from Section 4.3.

Homework

* TYPE YOUR SOLUTIONS TO exercises 1ab, 3, 5, and 18 from Section 4.3. (Notice the difference between exercises 3 and 4 - domain matters!)

4/27: ONLINE INSTRUCTION - FINALIZED

Due 5/1.

Required videos

* With your class text available in some form, open to Section 4.4 and watch these videos.
(1) Monotone Functions
(2) Monotone Functions Can Jump!
(3) Inverse Functions

Optional material

* Nothing really stands out in online videos.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 4.4 in the notes.

Homework

* TYPE YOUR SOLUTIONS TO exercises 1abc, 7, 10 from Section 4.4.

4/29: ONLINE INSTRUCTION - FINALIZED

Due 5/4.

Required videos

* With your class text available in some form, open to Section 4.5 and watch these videos.
(1) Learning About Gauges
(2) The Definition of a Gauge
(3) Forcing a Specific Tag
(4) Cousin's Lemma
(5) Using Cousin's Lemma
(6) Thinking About Gauges

Optional material

* Didn't find any videos out there.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 4.5 in the notes.

Homework

* TYPE YOUR SOLUTIONS TO exercises 1ac, 5, 6, 10, 13 from Section 4.5.

5/1: ONLINE INSTRUCTION - FINALIZED

Due 5/6.

Required videos

* With your class text available in some form, open to Section 4.6 and watch these videos.
(1) Coverings
(2) Open Covers and Compactness
(3) Compact Sets
(4) The Heine-Borel Theorem
(5) Full Covers

Optional material

* I found a couple of integrals, but two were over 30 minutes long, and the other had no visual, just audio. Google the gauge integral if you want to check them out.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 4.6 in the notes.

Homework

* TYPE YOUR SOLUTIONS TO exercises1acde, 3, 4, and 8 from this handout, which replaces the exercises on pages 235 and 236 of your text.

5/4: ONLINE INSTRUCTION - FINALIZED

Due 5/8.

Required videos

* With your class text available in some form, open to Section 5.1 and watch these videos. (I didn't realize that today was May the Fourth until after I made my videos, so I had to improvise a bit.)
(0) Introductory Videos
(1) The Definition of the Derivative
(2) Examples Written above the examples is the definition of the derivative: \(\displaystyle{\lim_{x\rightarrow c}\frac{f(x)-f(c)}{x-c}}\). Sorry - that got cut off for some reason.
(3) Continuity and Differentiability
(4) Some Familiar Rules
(5) The Chain Rule
(6) An Important Example

Optional material

* Lots of basic videos, not many at the level we need for this class.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 5.1 in the notes.
* Read all exercises in Section 5.1, especially 11.

Homework

* TYPE YOUR SOLUTIONS TO exercises 1ab, 2d, 7, 17.

5/6: ONLINE INSTRUCTION - FINALIZED

* Course evaluations are now live. Please complete a course evaluation before May 17.

Due 5/11.

Required videos

* With your class text available in some form, open to Sections 5.2-5.4 and watch these videos. I know there are more videos, but I promise that doesn't MEAN (sorry for the pun - one section is on the Mean Value Theorem) more homework. You do need to watch all of the videos, though, so budget a little extra time. Yes, one of these was actually recorded on a Wednesday. And no, I'm not sick.
(0) Why Three Sections in One Day? Please watch this.
(1) Rolle's Theorem
(2) The Mean Value Theorem
(3) Using the MVT
(4) Two Cool Examples
(5) The Most Amazing Theorem (The handwritten text should be "If \(f\) is differentiable on \([a,b]\), that does not imply that \(f'\) is continuous on \([a,b]\). For example, consider \(f(x):= x^2 \sin (1/x)\) on \([-1,1]\).")
(6) Taylor's Theorem
(7) Inequalities
(8) Taylor's Theorem and the Second Derivative Test

Optional material

* The Khan Academy describes the MVT intuitively.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 5.2 in the notes, skim Section 5.3, and read the parts of Section 5.4 that are referenced in the videos..

Homework

* TYPE YOUR SOLUTIONS TO exercises 1c, 11, and 16 from Section 5.2, and exercises 1b, 7bc from Section 5.4.

5/8: ONLINE INSTRUCTION - FINALIZED

I have decided to distribute your final exam on Wednesday of next week, since I'm not collecting that homework. For that reason, you may want to get a little ahead on homework this weekend, maybe even watch Monday's videos, so you can concentrate on the final after Wednesday.

Due 5/13.

Required videos

* With your class text available in some form, open to Section 6.1 and watch these videos.
(1) The Definition of Integrable
(2) Integrating a Few Functions
(3) Theorems About Integrable Functions
(4) Integrating an Unusual Function
(5) The Dirichlet Function and Thomae Function Revisited
(6) How to Show That a Function is Integrable

Optional material

* I didn't find much beyond the Calculus class level.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 6.1 in the notes.

Homework

* TYPE YOUR SOLUTIONS TO exercises 1acd, 5, and 9 from Section 6.1 in the notes.

5/11: ONLINE INSTRUCTION - FINALIZED

Because I will be distributing the final exam on Wednesday (I saw no need to wait until Friday since there is no new assignment on Wednesday), you may want to start this early and turn it in on Wednesday. I will post solutions on Friday. Solutions to Wednesday's homework have been posted already.

Due 5/15.

Required videos

* With your class text available in some form, open to Section 6.2 and watch these videos.
(1) Antiderivatives
(2) Proving the FTC
(3) Examples
(4) The Second Form of the FTC
(5) A Few Final Theorems

Optional material

* Not much online that I would recommend.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 6.2 in the notes.

Homework

* TYPE YOUR SOLUTIONS TO exercises 1abd, 2b, 9a, 11 from Section 6.2 in the notes.

5/13: ONLINE INSTRUCTION - FINALIZED

Exam 3 is now available on the Help Page.

You DO need to take the reading/video quiz, but you do not need to turn in solutions to homework. This homework will not be collected, but the material will be on the third exam. Solutions will be posted immediately.

Required videos

* With your class text available in some form, open to Section 6.3 and watch these videos.
(1) Why Improve the Riemann Integral?
(2) Getting to Know the Gauge Integral
(3) The Fundamental Theorem of Integral Calculus for Gauge Integrals
(4) Improper Gauge Integrals Aren't Improper

Optional material

* Not much available.

Reading assignment

* After viewing enough of the videos that you understand the material, read Section 6.3 in the notes.

Homework

* Solutions to exercises 1, 6, 7bd, 8 from Section 6.3 in the notes are already posted.

5/15: ONLINE INSTRUCTION - FINAL EXAM DISTRIBUTED ON WEDNESDAY, MAY 13. The exam will cover the material we studied from April 15 to the end of the course, with a few basic questions drawn from earlier in the course described on this handout.

This will be an open text exam. I will distribute the exam to you via the Help Page. Expect it to be available by 8:00 am. The exam will be due via email attachment by 10:00 am on May 19. You will work alone on this exam, and submit your work to me via email in a word-processed document. Word and pdf formats are acceptable. Specific guidelines on what resources you can use will be on the exam sheet..