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Alex Ginory, our LA, will be available for help, Tuesday and Thursday afternoons near DM 416. Make it a habit to attend !

I've grouped the class meetings into seven blocks of two weeks each. I will probably have to adjust some dates and content from time to time. I will do so as early as I can, and will announce any major changes in class. The HW lists are intended to be the minimal required to get by. You should do at least 50 per cent more.

How to read the tables: As you can see from the first table below, I expect to cover Chs 1 and 2 of Velleman's book in the first 2 lectures.Note: Homework 1 is due on 9/14, and goes through Ch 3. [So, HW1 includes Chs 1, 2 and 3]. Exam 1 is is 9/21, and also goes thru Ch 3 [for now]. I will usually post study advice, here on this page, about a week ahead of each exam, so check back often. Also, see my exam page, with practice exams and answer keys to your tests. Please contact me if you don't understand anything, or if you see a mistake.

Weeks 1-2

This can be a pretty hard course, especially after the first 3-4 weeks. To get a good start, buy both books and read Velleman Chs 1-3. Learn what's available here on my website such as the syllabus, policies, exam page, abbreviations, maybe some help pages. Get to know Alejandro, our Learning Assistant. Get to know your prof by coming to DM419B to visit in office hours, or email me and tell me a little about yourself.

In Weeks 1-2, we start to learn about proofs. Proofs in any subject are based on logic, and on definitions from that subject. Sets are the most basic subject, so we start with definitions about sets, and look at some proofs about sets. It's great if you learn a lot about sets here, but remember that the goal is an understanding of what a proof is, and how to write one. Think about that constantly. You might also practice with my tutorial - proofwriting. And, of course, do the HW.

Since sets are rather limited, we will also do a few proof examples from other subjects such as Calculus, even at the beginning. I want you to see that our proof strategies apply to any branch of math. By week 4 or 5, we will turn to some new subjects such as functions.

Weeks 3-4

In Ch 3, we study the main phrases of math and the main proof strategies that go with each one. We'll do a lot of fairly simple examples here. Notice the patterns. Eventually, you need to be able to pick a good proof strategy for such problems automatically. Optional - see my help page for a list of simple statements about sets that you can prove for additional practice.

Start studying and memorizing pages 376-379, which summarize the basic proof strategies.

Remember to have HW1 stapled and ready before class starts on 9/14.

HW2 will include some Calculus-based problems; I will post a link to them here soon. [For a preview, see problems A and B from my 2003 Limit HW,but I don't suggest starting them yet]. Read thru Ch3 and prepare for Exam 1 on 9/21. Optional - do the proof-writing drills on my help page.

Weeks 5-6

UPDATED Thurs 9/16/10 - Exam I will cover Chs 1 through 3, my lectures through 9/14 [eg one week before the exam, eg up to Ch 4.3], and HW 1. Know pages 376-379 (except for induction). Expect proofs similar to the exercises and examples in Chapter 3.1-3.7, perhaps involving sets (or integers or inequalities). Review examples from the lectures and the textbook. Prepare to use the definition of limit in proofs a bit above the Calculus I level (as in HW1). Be familiar with the notation and vocabulary we have used, well enough to use them in simple proofs [power set, x divides y, prime number, anti-symmetric relation, tautology, etc]

Remember that my exam page has old exams for a little more practice.

In chapters 4-7, we study some general topics, important in all areas of advanced mathematics. We have three goals - to learn these topics, to practice proof strategies and to develop good learning habits. This is where many students start to get lost. Some advice -

1) Do not fall behind !!! Use my office hours, study groups, email, etc to keep up.
2) Study each new definition in Ch 4-5, and relate it to a proof strategy on page 305.
3) Practice with lots of proofs and think them over afterwards. See my help pages for two detailed Ch 4 examples.
4) Imagine that we are building a perfect structure, that a major goal is to prevent errors, that every new statement requires a proof. (but in reality we may not have time for that).

HW 2 includes problems from Chs 4 and 5 listed in the usual place, and also these review problems HW2TF

on proof strategies and one limit exercise. I expect to give more limit excercises in HW 3 - if you want to practice more now, you can try these; 2003 Limit HW. There was an error in the HW2 list for Thurs 9/16 - omit Ch 4.5, but do problems from 4.6 [this has now been corrected above].

I have recently updated my "Help" page for this class, and the main page link to it. I have graded your HW2 and posted a partial answer key there. If you use the key, and find that it does help, let me know, and I'll probably post more [only one comment on these, so far this term].

Weeks 7-8

More topics from Velleman Chs 5-7. In Chapter 6, focus mainly on Ch 6.1 (induction). Look over Chs 6.2 and 6.3, but I do not expect to take any exam questions directly from those sections. Read 6.4 (Strong Induction) fairly carefully, including the proofs of theorems 6.4.1, 6.4.2 , 6.4.4 and 6.4.5. We proved that "Induction implies Strong Induction" in class. The converse is easier [but a bit tricky] - I have posted an explanation of that on my Help page.

Skip Ch 6.5 (Closures). We'll cover Chs 7.1 and 7.2, but not fully cover 7.3. Then, we will switch to the textbook by Shilov, refered to as "Book 2" or just "#2" below. We will also discuss number systems soon (not covered much in either book). We can do a lot or a little with that - let me know if you have a preference.

The FIU Math Club meets xxx pm, probably in xxx [if you know this info please share] . See Prof Yotov if you are interested in this, or in taking the Putnam Exam.

The DR / WI deadline is Oct 15, I think, but check this yourself. I don't usually get involved in this decision unless you ask my advice.

Notice that I have posted some special reading and HW for Tuesday 10/12 (above) but it is not due soon. I plan to post more asap, and have put a book by Morash on reserve in the library, in case you want to read more about the construction of our number systems. This is not required, as long as you understand my lectures and special readings on the subject. Also, I have recently posted some remarks about HW3 (and Strong Induction) on my Help page. I may post more remarks here and there before the exam, if I have time, and an internet connection.

Weeks 9-10

The rest of the course is based mainly on Shilov's book on Analysis ("Book #2"). Like most analysis books, it makes some simple assumptions about the real numbers (R), to move along faster. But we are supposed to study the construction of R in more detail, making as few assumptions as possible. We will need Cauchy sequences to do that. So, our plan is to follow Shilov until we are ready, and then "back up" and define R more carefully. I will provide notes, web pages about R, and/or special exercises at that time. For now, please follow the link(s) in the HW column of the tables, for reading in between our Velleman and Shilov periods.

Exam 2 will be based mainly on HWs 2-3, and Chs to 4-7 of Velleman. This is a lot of material, but if you've done the HW, you should not have to work too hard. Know the precise definitions of all the key terms, to use them in proofs (eg antisymmetric, partial order, function, onto, pre-image, congruence (mod 3), A~B, denumerable, etc, etc) Know the standard proofs, such as thms 5.2.5, 5.3.1, 6.4.2 , 6.4.4, 6.4.5 and 7.1.3. Know the limit proofs from HW 2 and the 2003 practice page [see the links above], though the most difficult of these probably won't be on Exam 2.

Note that HW 4 includes some problems based on the N, Z, Q notes and some from Shilov (Book #2).

I have put a book by Morash on reserve in the library, with a detailed construction of the systems N,Z,Q,R.

Weeks 11-12

I suggest taking Ch 3.3-3.7 and 3.9 more slowly and carefully than most sections. Here, you get an introduction to topology and sequences and practice with limit proofs [for sequences]. This takes some practice, so try to do a lot of problems. You can find more problems in any standard calculus book or Analysis book, if needed, and I may also write some for your next HW. Work through the textbook and lecture examples. This practice will pay off in the rest of this course and in any future analysis courses you take, such as MAA 4211.

We'll get to the definition of R soon, and then we can prove the properties that were assumed in Shilov Ch 1 (eg, that R is a complete ordered field). I will post an updated outline of all this soon and will fill in some of the proofs in class.

I've corrected a typo in the HW for Tues 11/2 and also edited the 10/26 HW to make it clearer. [11/2/10]. Eric gets a little extra credit for spotting this!

If you are interested in becoming a McNair Fellow, click here, and / or see me. If enough of you are interested, tell me, and we can invite Dr Banjoko to give us a presentation. In my opinion, this is a good opportunity for almost any math major [and many others]. Two math students recently won awards for their work done last year.

Week 13-14

Exam 3 will mainly cover topics related to HW 4-5; number systems, fields, Shilov Chs 1-3 [including open, closed and dense sets, but not including Cauchy sequences, or anything beyond that].Make sure you can do problems like HW 4-5 with confidence before the exam. I am willing to help you in any way I can with this. I'm posting a key to part of HW 5 [the problems for 11/2].

Know the proofs of Thms 1.71, 1.75, 1.81, 3.16, 3.22, 3.32c and d, 3.45, 3.52. I may give you some choice of proofs on the exam, but the more you know, the better.

Note that all work is due on 11/30 [late HW, regrading requests, excuses, etc].  It is very unlikely that I will grant incompletes - these are only for people who have last-minute (usually medical) emergencies.

Here are some comments and partial key to HW 6.

Notice that I have added some 11/30 HW, but these problems are not to be handed-in, they are just extra practice for the final exam. I may do some of these in class on Dec 2.

The final counts 30 points - about half will be on recent material (like an "Exam 4") and about half will be review. It cannot be dropped; see the policy page about this, about incompletes, etc. Be sure to do the Nov 30 "HW" problems (but don't hand them in). I plan to add more remarks about the final soon, including a short list of textbook proofs to know.

"Textbook" Proofs for the Final: 1) Every Cauchy seq is bounded and 2) converges (if the metric space is R); 3) the product of two Cauchy seqs is also Cauchy; 4) the real numbers are not countable. I will not ask you to construct R from Q, but might ask for some small step of that process (such as the definition of addition, or <, maybe with a short discussion).

Here is an old review page for this course. Please ignore anything there which is clearly out-dated (such as the list of proofs to know or topics we did not cover in Fall 2010). I have not had time to write a new review page, but would suggest reviewing the main definitions, notation and theorems if you haven't yet;

De Morgan's Laws, f(S), bijection, denumerable, anti-symmetric, well-defined, nested interval, BolzanoW, complete, lub, metric, interior point, dense, trichotomy, infinite, pre-image, bounded, etc.

Less mainstream, but maybe also worth review; lim inf, compact, Strong Induction, extended real number system, etc.

Final Exam : Tues. 12/7/10, 9:45am to 11:45am.

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