MAA 5228 and 4226 -- Introductory Analysis 1 (Fall 2015).

Instructor Of Record: Dr. J.D. Mireles James.
Office: SEC 262
Office Hours: Monday 4-5pm and Thursday 10-11am
(or by appointment).

Contact: The best way to reach me is by email at: jmirelesjames@fau.edu

Class Location: College of Education Bldg, Room 112. Class Time: MWF: 12:00pm-12:50pm

Textbook: Principles of Mathematical Analysis (Third Edition) by Walter Rudin.

Detailed Syllabus:

Professor Tomas Schonbek has taught the introductory analysis many times at FAU. The following comments and advice are taken from his course description: "Introductory Analysis is a two course sequence whose main purpose is to teach the basics of analysis in a rigorous and reasonably complete way. In our current setup, Part I can be called Calculus of one variable "done right." Part II covers the basics of Lebesgue integration. Together with the Introductory Algebra sequence it is supposed to prepare students for more advanced courses in mathematics. In our doctoral program, the qualifying exams are designed to test your readiness for taking advanced courses; an immediate consequence of doing well in the Analysis sequence is that you will also do well in the analysis qualifier. Of course, "doing well" does not mean simply getting a good grade, it means having understood the material, being able to prove the theorems, and being able to work most of the exercises on one's own. It may be useful to mention that writing carefully, expressing yourself in coherent sentences in the English language (both in writing and orally), understanding concepts, and proving theorems are essential ingredients of the course.

A few historical remarks may be in order. For many centuries, millennia even, mathematics was a relatively static science. Mathematicians studied properties of geometric figures, of numbers, and made many marvelous and beautiful discoveries: The existence of irrational numbers, Eudoxus' Theory of Proportions (which effectively resolved the crisis caused by the discovery of irrational numbers), Euclid's proof of the infinitude of the set of primes, Archimedes' sequences to compute π, his Method in general, Apollonius' work on conics, Diophantus on Number Theory, the introduction of zero and negative numbers by Indian mathematicians. The list could go on. But then, beginning in the sixteenth century with a few baby steps, more in the seventeenth century and finally, coming to fruition with the work of Newton and Leibniz, a sea change occurred; mathematics became dynamic. Functions appeared. In his Principia Newton studied what he called fluents (functions), and explained how to find the fluxion (derivative) of a fluent and, more interestingly, how to find the fluent given the fluxion (integration). From then on functions played a central role in classical mathematics, in analysis as well as in algebra, even though a first precise definition of the concept did not appear for quite a while. It can probably be traced to the work of Lejeune Dirichlet in the mid nineteenth century. Ironically enough, when rigor was taken to extremes in the twentieth century, it was found that the function, this most dynamical of objects, had to be defined in a static way as a set of pairs.

Not surprisingly, we will be spending a lot of time studying properties of functions, such as continuity, differentiability, integrability. We will, of course, also spend some time studying the sets on which these functions act, most particularly that extremely complicated and strange set known as the set of real numbers. We have to be rigorous. Newton, Euler, had a tremendous intuition into what they were doing. Most of us are not so fortunate and we need more guidance; rigor and precision provide that guidance. You can think of mathematics as a journey, an expedition, a trek. The journey at times can get quite difficult, but it is also very rewarding. Introductory Analysis and Introductory Algebra are the beginning of the journey, and a lot of time might be spent in doing the analog of fitness exercises, toughening you up for the difficulties that will come. You may find some of these exercises silly and boring, but there still is a point in doing them. You may find some of the going difficult but we assure you that if you stick to it, put a serious effort into mastering the basis."

There will be no make-ups. However, there will be ways in which you can make up for a missed exam or homework, as long as your grades are reasonably good.

Tutoring: For the first time ever the qualifying course will be supported by problem sessions in the Math Learning Center (MLC located at GS 211). (click here for the MLC page). These will be held every Wednesday starting August 26th, from 5:30-7:30pm in SE 212. The sessoins will be conducted by Yarema Boryshchak and Shane Kepley. Both Yarema and Shane are considered experts in this material and I have utmost faith in them. I strongly encourage all students to attend, even if you are feeling good about the material. Participation of all students will greatly improve the sessions. Also, no matter how well you are doing in the course you will benefit from discussing the material with Shand and Yarema. This is a fantastic resource for preparing early for the qualifying exam.

The Natural Numbers and a Little Set Theory:

---This pdf contains the first 20 pages of a book called "Numbers, sets, and axioms: the apparatus of mathematics." (view here).
The discussion of models satisfying the Peano axioms is very complete, i.e. there is a nice uniqueness result. I have not seen the rest of this book but my guess is that it would be very good for learning set theory.

--- A technical, but very well written introductoin to set theory can be found (here).
This is more set theory than we will need this year.

--- A set of notes which discusses the construction of the natural numbers from a set theoretic point of view is (here).

--- A short set of notes discussing the natural numbers from a purly axiomatic point of view is (here).
Note that the axioms are stated slightly differently than the way I did in class, but that they contain the same information.

--- A more complete set of notes discussing the axiomatic view of the natural numbers (is here).
Here the axioms are close to the form I stated in class.

The integers and rational numbers:

---A note about building the integers out of the natural numbers (view here).
The note goes over the construction we did in class but goes through all the details about extending addition, multiplication, the order relation, and subtraction to the new set.
Warning: the author uses the notation ``omega'' to denote the natural numbers, whereas in class we use a script ``N''.

---A note about building the rational numbers out of the integers (view here).
This note goes over the construction we talked about in class but goes into more detail about general algebraic fields.

The real numbers: axioms and construction

---Here is a note about the construction of the real numbers using Cauchy sequences. (view here). The treatment is very thorough, and similar to what I did in class. The only problem with these notes is that the least upper bound property is not established directly. (Though it can be proven from the completeness property, which the author does establish).

---Here is another note about about the construction of the reals. (view here). In these notes you can find the proof the the construction satisfies the least upper bound property.

---Here is a list stating the axioms for the real numbers. (view here).

---Here is a short note which gives a proof that Q does not have the least upper bound property. (view here).

---A short note which shows how to prove that the square root of 2 is a real number. The proof of course uses the least upper bound axiom. (view here).

## References:

I really like the book "Set Theory: An Intuitive Approach" by You-Feng Lin and Shwu-Yeng T. Lin. I think that this is a great book for students of analysis trying to learn `enough' set theory. Unfortunatly I don't have a pdf, but I have a copy in my office if you want to take a look.

Late homework will not be accepted. Homework accounts for 25% of your grade. While working with classmates and discussing homework with them is encouraged, the homework you hand in has to be your own work. The importance of working out the exercises ON YOUR OWN cannot be overemphasized. Of course, you may be stumped, and eventually you may have to ask for help for several of the exercises. But that should not happen until you have at least reached the point of suffering because the solution eludes you. If you ask for help before feeling any pain you are doing a serious disservice to yourself. You may pass the course with an A but you'll fail the analysis qualifier.

There will be two in class exams Each in class exam accounts for 20% of the grade with the final exam accounting for 30% of the grade. Class participation takes care of the last 5% of the grade.

Course outline   The plan is to cover Chapters 1 (briefly), 2, 3, 4, 5, 6, and 7 of Rudin's book; in other words, the following topics:

• The Real Number System. (two weeks)
• Metric Spaces (two weeks)
• Sequences and Series of points in a metric space (two weeks)
• Continunity (two weeks)
• Differentiability (two weeks)
• The Reimann integral (two weeks)
• Sequences and series of functions (two weeks)
I reserve the right to modify the schedule as we go. If we don't make it through all of this material in the first semester we will pick up where we left off on the Spring. Click here (or on the link below) for a detailed schedule, including the exam dates.

Grading   Your final grade will depend on homework, a midterm exam, class participation, and a comprehensive final exam. Homework will be assigned frequently. Roughly speaking an A is 90 percent or above, a B is 80-89 percent, a C is 70-79 percent a D is 60-69 percent and below a 60 percent is a failing grade. However I reserve the right to modify these definitions a little (in your favor) at the end of the semester.

Tentative Exam Schedule Exam 1 first week of october. Second exam third week of November.

Students at Florida Atlantic University are expected to maintain the highest ethical standards. Academic dishonesty is considered a serious breach of these ethical standards, because it interferes with the university mission to provide a high quality education in which no student enjoys an unfair advantage over any other. Academic dishonesty is also destructive of the university community, which is grounded in a system of mutual trust and places high value on personal integrity and individual responsibility. Harsh penalties are associated with academic dishonesty. For more information, see University Regulation 4.001.
Students with Disabilities:
In compliance with the Americans with Disabilities Act (ADA), students who require special accommodation due to a disability to properly execute coursework must register with the Office for Students with Disabilities (OSD) -- in Boca Raton, SU 133 (561-297-3880) and follow all OSD procedures.

Course Webpage: Check the homepage frequently as it will contain important course updates including homework assignments.

ASSIGNMENTS:

HW#1: TBA

ANNOUNCEMENTS AND SCHEDULE:

1) First class meeting is Monday, August 17th, 2015.

.
.
.

∞) Final Exam: April 26-th, 6:45pm--9:15pm.