We continue with 2.14.
I posted on the blog a proposition that can be helpful in dealing with infinite subsets of $\mathbb{R}.$
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 $a, b \geq 0.$
Assume $a < b.$ Prove that $a^n < b^n$ for all $n \in \mathbb{N}.$
If there exists $n \in \mathbb{N}$ such that $a^n < b^n,$ then $a < b.$
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.