Category Archives: Intermediate Analysis

Intermediate Analysis: Supremums and Infimums

Thursday morning, three students presented homework problems at the board. I followed this with a lecture introducing the concepts of supremum and infimum of a set. Because I got a little sidetracked, I did not quite make it to the Completeness Axiom.

What was the “rabbit chase” for this class? Well, at least a couple of students had commented to me, outside of class, on the difficulty they’ve had with recent homework assignments. They pointed out that they work and work and often can’t make any headway on a few of the proofs. They seemed a little discouraged by the fact that they need help from their professor on every assignment.

I took class time to reassure them that they are not alone. Just about everyone in the class is going through the same thing. I pointed out the fact that I was in their place not that long ago. In fact, since I also did my undergrad here at Wayland, I was almost exactly in their place. I recounted tales of my discouragement as well as the fact that I also spent time in my professor’s office get help on almost every assignment. I was largely motivated by a reader of this blog, who is also a blogger I read regularly. He has recently made the point that the students can gain a new level of insight to a subject by seeing the learning process that the instructor, themselves went through to understand certain concepts.

In my mind, although some may disagree, it does not get any harder as an undergrad than a senior level mathematics course. There are many courses that require as much “work” as a course like this, but I can’t think of too many that require such an high level of abstract and critical thinking. I’ve yet to be convinced otherwise. However, I’ll admit that my undergraduate Physical Chemistry class may have been close.

Next time, we’ll finally cover the completeness axiom.

Intermediate Analysis: Wayne’s World

What does “Wayne’s World” have to do with Real Analysis?

Good question. I inadvertently structured my lecture in such a way that it reminded me of the old saturday-night-live-turned-box-office-money-maker. Both in the sketch and the movies, Wayne and Garth would broadcast their rock-n-roll show from Wayne’s basement. Anyways, whenever they would do a dream segment or a “back in time” segment, they would mimic the “fuzzy” screen effect and dreamy music my waving their hands frantically and saying something to the effect of “doodle-oodle-oodle-…” (DISCLAIMER: By referring to Wayne’s World, I in no way condone some of the associated raunchy humor, even though I probably laughed at some of it – shame on me.)

During class, I started a section of material and after beginning a proof, realized that we had not yet covered some essential material. Thus, we had to do a dream sequence, much like Wayne’s World or any other movie where you see the end of the story and then go back in time to see how you got there. How about that for interesting use of lecture style?

I had begun the definition of an \varepsilon-neighborhood and in trying to explain exactly what it was, I realized that we had some left some important results of the absolute value had not been established. Those results are well-known facts but we have emphasized the importance of fundamental principles in this class. We use (almost) nothing without first establishing the facts we need through proofs or use of axioms and definitions.

I finished with a few examples demonstrating sets that have the varying properties:
(i) Every point in the set has a an \varepsilon-neighborhood that is contained in the set.
(ii) There is a point in the set that, every \varepsilon-neighborhood of that point contains both elements of the set and elements in its complement.

Next time, we’ll give those concepts names after we establish the identifying property of the reals, namely, the completeness axiom. We also have homework due so there will be some oral presentations.

Intermediate Analysis: More order principles and absolute values

We had no homework assignments due on Thursday, so I dove right into lecture. We proved a few more results following from our ordering of the reals. Upon completing these theorems, we were then able to do the basic sorts of solving of inequalities we teach in our lowest level of algebra. It is interesting that we get to one of our final senior undergraduate courses and start doing things we cover in the earliest mathematics courses, but this time, we’ve built our understanding of these rules from the ground up. Have no fear, analysis students, we will go much, much beyond those mathematical tools we used intermediate algebra.

Next, we defined the concept of absolute value and some simple results of that definition. The next big thing after that was to use absolute values to create \varepsilon-neighborhoods which will provide us the necessary structures to define open and closed sets, cluster points, limits, etc.

Before we get to those concepts, however, we will divert to another low-level algebra topic of solving absolute value equations and inequalities. That will happen next time at the very beginning of class.

Intermediate Analysis: Order Properties

During this morning’s class I surprised the students by letting them know that today was my turn to answer homework questions at the board. As I have mentioned in previous posts, the students are being randomly selected to present some of the homework problems at the board. For many of them this is a considerably stressful event and with today’s assignment being particularly challenging, I noticed that many were very relieved when I offered to answer some of the questions at the board. Plus, it saves me needing to grade those particular problems.

For the lecture portion of today’s class, we went over the order properties of the Real Numbers. I recall the first time I learned this material and was actually quite perplexed by the definition of the set of positive numbers. Of course, I had an intuitive concept of what I meant when I said a number was positive but that was dependent on an ordering that was "inherited" from concept of the number line. Instead of starting with a number line or a given ordering, we simply define a set P that has the following properties:

  1. \forall \ a, b \in P, \ \ a+b \in P
  2. \forall \ a, b \in P, \ \ a\cdot b \in P
  3. \forall \ a \in \mathbb{R}, exactly one of the following hold:
    (i) a \in P
    (ii) -a \in P
    (iii)  a=0

This set, which we define to be the positive real numbers, provides a complete ordering on the set of reals. One of the students asked a question that I remember asking myself. Why can’t the set P be any other set, say, the negatives. The answer is that (ii) is violated when we consider the negative reals.

Next time, we begin considering the topology of the reals, looking at \varepsilon-neighborhoods, open and closed sets, cluster points, etc.

Intermediate Analysis: Real Numbers

On Thursday, we began the chapter on Real Numbers. We are taking the axiomatic approach to defining the set of real numbers. I demonstrated the difficulty of defining Real Numbers in the techniques that have been typically used in their earlier classes, such as by decimal expansion or as points on the real number line. One of the things we discussed was the ambiguity of decimal expansions, for example

0.99999999\ldots = 1

I also alluded to the method of actually constructing Real Numbers through the method of Dedekind cuts but I have chosen to follow the text by describing the properties that uniquely define the set.

We began with with the field axioms and proved all many of the common results that follow from them, such as the fact that the product of the additive identity with any other real number must be the additive identity. This is the reason that the additive identity cannot have a multiplicative inverse. It was a fairly difficult concept to grasp for the students that we are not necessarily talking about “numbers” at this point but simply a set of objects that obey a certain set of properties. When I write 1 or 0 on the board, it is very helpful to keep in mind that these aren’t just the typical numbers that we’ve always thought them to be, but instead they are symbols used to represent the multiplicative identity and the additive identity, respectively. They are simply the objects that when you multiply or add to another element of the set of real numbers, you get what you started with. I find it very exciting to be developing their ability to think abstractly. I really do love this course, even if a few of them seem to get glossy-eyed at times.

Nevertheless, I am a little saddened by the fact that this is the last undergrad math course for many of them. I know at least a few should end up at graduate school but some may not have any other math courses beyond this point. Plus, most of them have heard all my jokes and stories so they can quote the punchline before I finish setting them up. I sometimes feel like a stand-up comedian with an audience that’s heard all my material. I guess I really need to expand my repertoire.

Intermediate Analysis: Cardinality

During I.A. on Tuesday (Holy Cow 7:25 am is early to start a class) two students presented problems at the board. I really enjoy having this part of the class. All of these students have taken courses from a colleague of mine. She has made oral presentation of homework a regular part of her upper level courses. So, they are all used to it, but it is fairly new to me. I enjoy hearing the thought processes of the students as they assimilate this knowledge. I even developed Rubric to grade these presentations. Don’t I sound like an education geek? (By the way, I was never formally trained to teach.)

We then covered a few more results regarding countability, such as categorizations of countability, the fact that the countable union of countable sets is countable, and the countability of the rationals. We then diverted from the text a bit and introduced the concept of cardinality and cardinal numbers. I introduced them to aleph-null as the “size” of the natural numbers and the smallest of all transfinite cardinal numbers. We, then, proved that the “size” of the reals is strictly larger than the set of natural numbers, i.e., the Reals are uncountable. I ran out of time before covering Cantor’s Theorem. It is one of my favorite proofs in this class so there’ll be an entry in the next day or two over just that. I’d describe it as “slick.”

Next time, we finally finish Chapter 1 and begin covering The Real Numbers. What the heck are they anyway? We’ll using the axiomatic approach.

Intermediate Analysis: Countability

CountableIn Intermediate Analysis this morning, three students each went to the board to present homework problems on basics of sets and mappings. This is first time that I have incorporated the oral presentation of homework as a part of an upper level mathematics course. It was one of my least favorite aspects of the graduate courses that I took and yet I realize just how formative it was in the development of my understanding for those courses. I have a stronger grasp of those courses in which such presentation was an essential portion of the course.

The lecture for today covered the concepts of denumerable (countably infinite) and countable sets. We proved that some basic sets are denumerable, such as the natural numbers, even numbers, odd numbers, the cartesian product of countable sets, . . . We did not quite get to the proof that the reals are uncountable but that should happen at the beginning of the next class. I’ve decided to add some additional information to the course that is not included in the text at this point. We are going to discuss cardinal numbers and their ordering. We’ll broach the subject of the Continuum Hypothesis in the next class.

I can recall the first time I encountered the idea that the concepts of infinity with respect to infinite limits and limits at infinity are distinct from the cardinality, or “size”, of infinite sets. It blew my mind when I first learned that the sets  \{ 1, 2, 3, 4, \ldots \} and  \{2, 4, 6, 8, \ldots\} are, in fact, the same size, even though only half the first set appears in the second set. But they are the same size of “infinity” while the number of real numbers is larger. That is, it is a strictly larger infinity than the number of counting numbers. All of this good stuff comes next time in class.

By the way, I have to get better about not ending class in the middle of a proof. A more judicious use of my time is needed. (Like that will ever happen). I am already doing poor on my dismissal of class on time. Although, the latest I’ve let them out is 3 minutes late. I will work on that as well.