Spring 2011 MATH 312: Proofs in Elementary Analysis Branko Ćurgus

Monday, June 6, 2011

• Somewhat late, here is Wes's perfect Assignment 1.
• Here are solutions of some problems from Assignment 2.

Tuesday, May 31, 2011

• I am posting the proof of the "zero theorem" that I did today. I made the proof 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".
• On Thursday I will prove the "max theorem". Here is the proof. This proof is also self-contained.

Thursday, May 26, 2011

Monday, May 23, 2011

• Exercises done in 2.8: 2,3,6,12,14,15, 18 (not done, but easy)
• Exercises done in 2.9: 7.
• Exercises done in 2.10: 3, 6.
• Exercises done in 2.11: 1, 2, 4, 5.
• Exercises done in 2.12: 1, 2, 3, 5, 7, 8, 18.
• Exercises done in 2.13: 3, 4, 5, 6, 7, 8, 9 (not done but you should know it), 12.
• Exercises done in 2.14: 2 (I did a simpler example 1/n), 4, 5 (done with the specific example (0,1)).

Friday, May 20, 2011

• Exercises to do from 2.14: any.

Monday, May 16, 2011

• Exercises to do from 2.10: 2.10.3, 2.10.4, 2.10.5.
• Exercises to do from 2.12: 2.12.2, 2.12.9, 2.12.19.
• I did 2.12.1, 2.12.3, 2.12.5, 2.12.7, 2.12.8, 2.12.10, 2.12.18.
• Exercises to do from 2.13: 2.13.3, 2.13.4, 2.13.6, 2.13.7, 2.13.8, 2.13.9, 2.13.10, 2.13.11.
• I did: 2.13.5, 2.13.12.

Friday, May 7, 2011

• There was a typo in the assignment that I handed out. The exercise mentioned in Problem 4 should be Exercise 2.9.7. The corrected version you can read here.

• The current positive presentation scores in decreasing order are 25, 14, 14, 13, 12, 6, 6, 6, 6, 5, 3, 2, 2, 2 making the class average 6.44 with the standard deviation 6.55.

Saturday, April 30, 2011

• Read Section 2.8. Understand the definitions and all the statements. This section consists of three parts: the statements about finite sets, the statements about ${\mathbb N}$ and the statements about countable sets. You should do at least one proof in each part.

• The most important exercise about finite sets is: 2.8.6. (5 points for a successful presentation). Mathematical induction is a natural way to proceed here. It is essential to be specific what is the statement $P(n)$ that is being proved.

• Notice that I proved Exercise 2.8.15: Every nonempty subset of ${\mathbb N}$ has a minimum.

• Abstract proofs about countable sets are somewhat technical. Using what is proved in the notes, you should be able to do 2.8.21. As always, it is essential to distinguish between green and red parts of the proof. Then one can develop a strategy to construct red parts using green.

• The current positive presentation scores in decreasing order are 20, 14, 12, 11, 10, 6, 6, 6, 6, 3, 2, 2, 2, 2 making the class average 5.67 with the standard deviation 5.54. I included all the recent blogs.

Tuesday, April 26, 2011

• The current presentation scores in decreasing order are 20, 14, 11, 11, 10, 6, 6, 5, 3, 3, 2, 2, 2, 1 making the class average 5.05 with the standard deviation 5.53

Friday, April 22, 2011

• Read Section 2.6. Do exercises from 2.7.
• Credit is available for posting of solutions to 2.5.18, 2.5.17, 2.5.16, 2.5.13.
• The current presentation scores in decreasing order are 16, 11, 9, 8, 7, 6, 5, 5, 3, 3, 2, 1 making the class average 4 with the standard deviation 4.45

Tuesday, April 19, 2011

• I think that all exercises in 2.2. and 2.3. are now done. Please post 2.3.6.
• Please post solutions to 2.4.5., 2.4.6. 2.4.7. for points. I did 2.4.10 in class.
• Please post 2.5.7, 2.5.8 for credit. For Thursday prepare 2.5.10 (together with 2.5.11), 2.5.13, 2.5.14, 2.5.15, 2.5.16, 2.5.17, 2.5.18.

Tuesday, April 12, 2011

• The following exercise should be added in 2.2:

Exercise 2.2.11. Let $\alpha \in {\mathbb R}.$ Prove that $\alpha < x$ for all $x > 0$ implies $\alpha \leq 0.$ The proof is simple. Just state the contrapositive. If $\alpha > 0,$ then there exists $x > 0$ such that $\alpha \geq x.$ The contrapositive is clearly true since we can take $x = \alpha.$

• Robert used this exercise correctly in his proof.
• The proof of Exercise 2.3.5 can be simplified using this new exercise.
• A slightly stronger version of Exercise 2.2.11 is also true:

Exercise 2.2.11a. Let $\alpha \in {\mathbb R}.$ Prove that $\alpha \leq x$ for all $x > 0$ implies $\alpha \leq 0.$ The proof is again simple. Consider the contrapositive. If $\alpha > 0,$ then there exists $x > 0$ such that $\alpha > x.$ Again, the contrapositive is true since we can take $x = \alpha/2.$

Monday, April 11, 2011

• We did 2.2.5 (a post is needed), 2.2.8 (two proofs are already posted, each earned one point), 2.2.9 (a post is needed).
• Please post proofs of 2.2.7 and 2.2.10 for credit.
• Tomorrow you can present 2.3.3, 2.3.4, 2.3.6, 2,3,7.
• The current presentation scores in decreasing order are 8,5,5,4,3,3,2,1,1, making the class average 1.60

Sunday, April 10, 2011

• A comment on a post on Friday: I overlooked that 2.2.2(d) and (f) are already proved in the notes.

Friday, April 8, 2011

• I posted solutions to 2.1.2 (b) and 2.1.2 (c) on the blog.
• 2.1.2(d) and (i) need proofs to be posted. You will earn points if you do it.
• 2.2.2(d) and (f) need proofs to be posted. You will earn points if you do it.
• Students who presented should post their solutions on the blog. If you have problems with LaTeX, write what you can; I will correct your LaTeX.
• I proved 2.2.4 in class today. Also, a posting of a solution will earn you credit. For Monday do 2.2.5, 2.2.7, 2.2.8, 2.2.9, 2.2.10.

Thursday, March 31, 2011

• I invite you all to post a comment to my post Writing math with LaTeX. Try to post a formula that made an impression on you from some other class. But have in mind that formulas in comments do not display well in Internet Explorer. All works well in Firefox and Chrome. Please test other browsers and let me know.
• LaTeX was written by Laslie Lamport. I wanted to know more about him, so I visited his website. There I found a quote which I think is relevant to all education, not just education of computer engineers: "Education is not the accumulation of facts. It matters little what a student knows after taking a course. What matters is what the student is able to do after taking the course. I've seldom met engineers who were hampered by not knowing facts about concurrency. I've met quite a few who lacked the basic skills they needed to think clearly about what they were doing." This is a quote from Laslie Lamport's paper "Teaching concurrency" (the paper number 167 on his writings website). The bold face is mine. How do we teach skills that one needs to think clearly? I believe that each mathematical problem is an exercise in thinking clearly about what you are doing. But, you have to make it so. And, I hope that I can help in that.

Wednesday, March 30, 2011

• Here is a link to our blog
• Today I posted "Writing math with LaTeX" on our blog. You can practice posting formulas on the blog as comments to this posting. Please let me know if you have problems.
• Here is a file with basic LaTeX symbols.

Tuesday, March 29, 2011