Fall 2009 MATH 312: Proofs in Elementary Analysis Branko Ćurgus

Friday, December 4, 2009

• I am posting the proofs presented yesterday and today. I made the proofs completely self-contained. That is I am not citing any exercises from the notes. I just use the definition of continuity. Here is the "zero theorem". Here is the "max theorem".

Thursday, December 3, 2009

• Here is a sample TeX file showing a simple way to adjust margins.
• The current class scores are:
28, 27, 20, 20, 18, 17, 16, 16, 13, 11.
The average is 18.6 and the standard deviation is 5.18.
The corresponding (tentative) percentage scores based on the formula given in the information sheet are
96, 93, 74, 74, 69, 66, 63, 63, 55, 50
These scores are calculated based on "the class average" of 70 and "the class standard deviation" of 14.16. (For the definitions of these quantities see the information sheet.) Have in mind that the grades for Assignment 2 are missing. Here "the class average" and "the class standard deviation" are calculated based on E1, E2 and A1. But, I do not expect that they will change much after A2 is graded.
• As an incentive for you to write your assignment in LaTeX I am posting a TeX file with Assignment 2 problems and with some hints for solutions. I might extend the hints later on. Just in case, here is a pdf file as well. Happy LaTeXing.

Monday, November 30, 2009

• These are exercises that we should do this week: 4.4.1, 4.4.2, 4.4.3, 4.6.1, 4.6.2, 4.6.3, 4.6.4, 4.6.5, 4.6.6.

Wednesday, November 25, 2009

• This is what I wrote during the exam.

Thursday, November 24, 2009

Monday, November 23, 2009

• Remaining exercises: 3.13.12, 3.13.13, 3.14.3, 3.14.4, 3.14.5, 3.15.2.

Friday, November 20, 2009

• I will post few problems from the second assignment on the blog during the weekend.
• The current class scores are:
25, 24, 18, 17, 17, 17, 16, 13, 11, 10.
The average is 16.8 and the standard deviation is 4.643.

Thursday, November 19, 2009

• Here is the list of available exercises ordered by their priority: 3.13.12, 3.13.13, 3.15.1, 3.15.2, with the new ones 3.14.2, 3.14.3, 3.14.4. 3.14.5, 3.14.6.

Tuesday, November 17, 2009

• Here is the list of available exercises ordered by their priority: 3.11.4, 3.13.12, 3.13.13, 3.13.6 (with 3.13.5), 3.15.1, 3.15.2, 3.8.5, 3.8.7 (together with 3.8.6).

Monday, November 16, 2009

• Please check if there are any "cheap points" among 3.8.3, 3.8.5, 3.8.7 (together with 3.8.6), 3.9.2, 3.10.1, 3.11.4, 3.13.5, 3.13.6, 3.13.12, 3.13.13. These are the exercises for this week.

Friday, November 13, 2009

• Chapter 3. Read 3.1, 3.2, 3.3, 3.4 and 3.5. Most of this material should be familiar from Math 226. The following exercises are available for presentation: 3.4.6, 3.4.7, 3.5.1, 3.7.1.
• The current class scores are:
22,19,17,16,16,14,13,12,10,3,0.
The average is 12.91 and the standard deviation is 6.259.

Tuesday, November 10, 2009

• We continue with 2.14.

• I posted on the blog a proposition that can be helpful in dealing with infinite subsets of $\mathbb{R}.$

• Now I offer 3 points for a complete blogged proof of 2.12.18. A "complete" means that you have to prove any unproved exercise that you use in your proof. (Now this is done!)

Monday, November 9, 2009

• Exercises 2.13.4, 2.13.5 and 2.13.6 are done. Notice that we proved that Exercise 2.13.5 is equivalent to the Completeness Axiom. In the same way, the Completeness is equivalent to 2.13.6. I hope that now Exercises 2.13.7 and 2.13.8 are "cheap points".
• We are moving on to 2.14. All exercise are available for presentation. Please pay attention to details.
• Notice that Exercise 2.12.13 is posted on the blog. Now I offer 2 points for blogged complete proof of 2.12.18.

Friday, November 6, 2009

• We are moving on to 2.13. The following exercises have priority: 2.13.4, 2.13.5, 2.13.6, 2.13.7, 2.13.11, 2.13.13.
• The remaining exercises from 2.12 can be posted on the blog. Here I list them in the order of importance: 2.12.19, 2.12.18 (can be proved using 2.12.5; if you use 2.12.17, you should prove it as well), 2.12.8, 2.12.9, 2.12.10, 2.12.11, 2.12.13, 2.12.14, 2.12.16.
• The current class scores are:
19,16,14,12,11,11,10,10,10,3,0.
The average is 10.55 and the standard deviation is 5.088.

Friday, October 30, 2009

• I posted some hints for 2.10 on the blog. Also, if you have questions about the assignment you can post them on the blog. I will respond.

Thursday, October 29, 2009

• Exercises for Friday 2.10.1, 2.10.2, 2.10.3, 2.10.4 and 2.10.5.

Friday, October 23, 2009

• Today we did 2.8.11, 2.8.12 and 2.8.16. The exam is on Monday. It covers 2.1 through 2.8. While you review for the exam, you could blog the proofs that you did in class, if you have not already done so.
• The current class scores are:
12,12,11,10,10,10,10,8,8,6,4,3,2,0.
The average is 7.571 and the standard deviation is 3.774.

Thursday, October 22, 2009

• Today we did 2.8.9, 2.8.10, 2.8.15 and 2.8.17. The remaining problems are 2.8.11, 2.8.12, 2.8.16. Please read the proofs given in the notes. As an incentive I will give you points if you find mistakes.
• Brett adjusted the previous blog template to accommodate long formulas. So, we are back to the old look. Thank you Brett.

Wednesday, October 21, 2009

• Here is a link to a webpage with few "proofs" that Malcolm mentioned yesterday. The proof which uses mathematical induction is particularly relevant.

Tuesday, October 20, 2009

• The only remaining problem in 2.7 is 2.7.3. We can leave it for the blog. It should be reformulated in two parts. In both parts below we assume $a, b \geq 0.$

1. Assume $a < b.$ Prove that $a^n < b^n$ for all $n \in \mathbb{N}.$

2. If there exists $n \in \mathbb{N}$ such that $a^n < b^n,$ then $a < b.$

• We discussed 2.8.8, 2.8.13, 2.8.17 and 2.8.21. I will post on the blog a summary of the discussion. There remain 2.8.9, 2.8.10, 2.8.11, 2.8.15 and 2.8.16. There is at least one super easy problem here. Also, two of the problems are just contrapositives of each other.
• Since some of the formulas in the recent blogs were long, I had to change the template. I liked the previous template better, but I do not know how to adopt the size of the post to longer formulas.
• The current class scores are:
10,9,9,8,8,8,7,7,6,5,4,2,2,0.
The average is 6.071 and the standard deviation is 2.939.

Friday, October 16, 2009

• Today we did Exercises 2.7.5, 2.7.9 and 2.8.17. Unfortunately our blog is not rendering the LaTeX code today. Hopefully this will change soon.

• The current class scores are:
9,8,8,8,7,7,7,7,6,5,4,2,2,0.
The average is 5.714 and the standard deviation is 2.63.

Thursday, October 15, 2009

• Today I talked about Section 2.8. I provided a proof of Exercise 2.8.8. Many proofs in this section essentially use previous proofs. In fact, this is often the case in math: You need to recognize a connection of what you are proving to what has been proved before. In Section 2.8 some of the exercises are very easy consequences of the previous stuff. Pay attention! And score some easy points.

• Few remarks about our blog: Exercise 2.5.11 is blogged. The idea from this proof can be used to give a short proof of 2.5.10. Exercise 2.5.16 is a consequence of exercises preceding it. These can be some easy points. A correct blogged solution to the exercise posted here on October 9th, fourth bullet, will get 2 points.
• Please comment on the blog if you have questions or see weaknesses in blogged proofs.

Tuesday, October 13, 2009

• Today we did Exercises 2.7.4, 2.7.5, 2.7.8. Please post these exercises on the blog. Exercises are being blogged at a good rate. Please keep up; both, posting and studying. Please comment if you have questions or see weaknesses in blogged proofs.

• Exercises 2.5.11, 2.5.12, 2.5.15 are "blog only." Old "blog only" exercises are 2.5.5 and 2.5.18.
• For Thursday we have 2.7.3, 2.7.9, 2.7.10. Also, please read Section 2.8. There are several proved theorems and solved exercises in this section. Based on these theorems and exercises you can do Exercises 2.8.8 through 2.8.17.

• The exercise offered on October 9 is still available. The proof which is blogged has a weakness: It uses $\sqrt{2}.$ The existence of this number has not been proved yet. In fact the existence of $\sqrt{2}$ cannot be proved based on the axioms that we introduced so far.

• The current class scores are: 8,7,7,7,7,7,6,5,5,5,2,2,2,0. The average is 5 and the standard deviation is 2.42.

Monday, October 12, 2009

• Today we did Exercises 2.4.6, 2.5.13, 2.5.15, and 2.5.17. Please post these exercises on the blog. In fact, several exercises were posted today. Thanks to the authors. Please comment if you have questions or see weaknesses in blogged proofs.

• On Tuesday the priority will be given to 2.7.3, 2.7.4, 2.7.5, 2.7.7, 2,7,8, 2.7.9, 2.7.10.

Friday, October 9, 2009

• Today we did Exercises 2.4.11, 2.5.6, 2.5.7, and 2.5.12. Please post these exercises on the blog. And please post the old ones as well. Our blog backlog is growing. If you encounter problems (for example the blog does not accept what you want to post) email me the file and I will fix the errors that I see.

• The following exercises are now "blog only" for 1 point each for a clear clean proof posted on the blog: 2.5.3 (blogged), 2.5.4 (blogged), 2.5.5 (These are similar to what we did in class. It seems to me that the only way to prove them is to consider several cases.) 2.5.8 (This is a direct consequence of Exercise  2.5.6(a); there is a short proof.) and 2.5.18.

• The next exercises are 2.4.6 and 2.5.10 (this goes together with 2.5.11), 2.5.13, 2.5.14, 2.5.15, 2.5.16, 2.5.17.

• Since we did several easy problems involving the minimum and the maximum, I am offering the following exercise for credit:

Consider the set   $A = \bigl\{ x \in \mathbb{R} : x > 0 \ \text{and} \ x^2 > 2 \bigr\}.$   Prove that $A$ does not have a minimum.

There is a proof of this statement in Section 2.11. Please do not read this proof. Your proof must be different.

• Read Section 2.6. There are no exercises to be presented but this section is essential for the further development of the material. Next exercises are in Section 2.7. Section 2.8 is mostly reading with few exercises.

• The current class scores are: 5, 5, 5, 5, 5, 5, 5, 4, 3, 3, 2, 2, 0, 0. The average is 3.5 and the standard deviation is 1.8028.

Thursday, October 8, 2009

• Today we did Exercises 2.4.8, 2.4.5, 2.4.4 and 2.4.13. Please post these exercises on the blog. Several old exercises were posted today. Thanks to the authors. Please keep on posting.

• If you have problems posting stop by my office or email me. Sometimes it is easy to solve problems. Do not copy and paste from MS Word to the blog. You can write in Notepad and then copy and paste. There is a better way to do this. Here is a blank TeX file called SAMPLE.TeX. Right-click on the underlined word "Here"; in the pop-up menu that appears, your browser will offer you to save the file in your directory. Save the file as SAMPLE.TeX. After saving the file you can open it with WinEdt. You will find WinEdt on all campus computers in
Start -> All Programs -> Math Applications -> WinEdt.
Open WinEdt first, then open SAMPLE.TeX from WinEdt. Then, in WinEdt you can write your solution. There is a short text in SAMPLE.TeX. You can use this small file to see how LaTeX works:
1. On the toolbar manu in WinEdt identify two icons: LaTeX and DVI (one on the right, without an arrow).
2. Click on the LaTeX icon. LaTeX will compile the file and create a dvi file. If there are LaTeX errors in the file you will be notified. If all goes well that means that your LaTeX code is ok.
3. The previous step created a dvi file. To see this file click on the DVI icon. in a new window you will see your file with nicely displayed formulas.
Now you can "save as" SAMPLE.TeX with a different name and write your solution. After you wrote few formulas, repeat steps 1, 2, 3, from above: click the LaTeX icon, if all goes well click the DVI icon. When your dvi file looks good you can copy and paste your TeX file to the blog and save your work as a TeX file for further reference.
• One warning: the blog does not like the strictly less sign < next to a letter. So, x<a will always lead to problems. Therefore, please write the symbols separated with spaces: x < a. In fact, it should be always within  signs: $x < a.$
• The next exercises are 2.4.6, 2.4.11 and 2.5.5, 2.5.6, 2.5.7, 2.5.8, 2.5.10 (this goes together with 2.5.11), 2.5.12, 2.5.13, 2.5.14, 2.5.15, 2.5.16, 2.5.17, 2.5.18.

Tuesday, October 6, 2009

• Today we did Exercises 2.3.4, 2.3.7 and 2.3.8. Please post these exercises on the blog. James wrote Ex. 2.4.4 but we did not have time to discuss it. It would be nice if he could post it (for points, of course), then we can discuss it in class. Several old exercises were posted today. Thanks to the authors.

• The next exercises are 2.4.5, 2.4.6, 2.4.8, 2.4.11, 2.4.13 and 2.5.5, 2.5.6, 2.5.7, 2.5.8, 2.5.10 (this goes together with 2.5.11), 2.5.12, 2.5.13, 2.5.14, 2.5.15, 2.5.16, 2.5.17, 2.5.18.

Monday, October 5, 2009

• Today we did Exercises 2.2.5 and 2.2.3, in this order, and 2.3.5. These exercises and the old ones 2.2.8 and 2.2.9 need to be posted on the blog.

• For tomorrow we have Exercises 2.3.4, 2.3.7, 2.3.8. These are very similar to each other; proved by a somewhat repetitive argument; but pay attention to detail in your proof; pay attention to syntax. The next exercises are 2.4.5, 2.4.6, 2.4.8, 2.4.11, 2.4.13.

• You can practice posting formulas on the blog as comments in the posting of September 29, "Using LaTeX on this blog". I will post some problems from the notes there.

Thursday, October 1, 2009

• Here is the list of all axioms of $\mathbb{R}.$ Just to remind you: AE, AA, AC, AZ, AO, ME, MA, MC, MO, MR, DL, OE, OT, OA, OM, and the most important one, CA, the Completeness Axiom which will be introduced later.

• Today we did 2.2.8, 2.2.9 and 2.2.10 (already posted on the blog). Exercise 2.2.3 is not difficult. The key is to understand that the numbers $a$ and $b$ are green (that is given) and the number $c$ is red. So, you have to make $c$ using $a$ and $b$ and other green numbers mentioned earlier. Ooops, now I realize that for this you will need Definition 2.2.4 and Exercise 2.2.5. So, for tomorrow two easy ones, Exercise 2.2.5 and Exercise 2.2.3, in this order. Also, Exercises 2.3.4, 2.3.5, 2.3.7, 2.3.8.

Tuesday, September 29, 2009

• Here is a link to our blog
• Today we did Exercises 2.2.2 (b), (g), and 2.2.7. I invite the presenters to write their proofs (improved) on the blog. You should try to prove Exercises 2.2.3, 2.2.8, 2.2.9 and 2.2.10. Again, the point is on rigorous justifications of all the steps in your proof. You should also look at Exercises 2.3.4, 2.3.5, 2.3.7, 2.3.8.
• In my Math 224 we use the distance formula for two points in space. That formula is proved using the Pythagorean theorem. I wanted my students to see a proof of the Pythagorean theorem so I made a visual proof in Mathematica. Here is a link to that proof. This is a proof in geometry, and I did not give all the details, but I am hoping that you can fill in the details.

Monday, September 28, 2009

• Today I will talk about the axioms of the real numbers. I will reprove some of the statements that are proved in the notes. You should try to prove Exercises 2.2.2 (b), (g), 2.2.3, 2.2.7, 2.2.8 and 2.2.9. Here the point is on rigorous justifications of all the steps in your proof.

Thursday, September 24, 2009