References are to the text by Kenneth A. Ross (corrected 12th printing, 2000), except when otherwise specified.

You may move to a particular week by clicking on its number: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
 

Read this file. If you have any question about it, ask me.

Read Section 1.1 (on the set of natural numbers). Your objective is to understand proofs by induction, and learn how to write such proofs.

Read Section 1.2. The problem is, Given a polynomial with integer coefficients, find all of its rational zeros. There is a Theorem that provides a solution. Your objective is to understand the Theorem, know its proof, and be able to apply it.

Read Section 1.3, which describes the familiar algebraic structures in our numbers systems. I'll not talk much about these in class.

Algebraic Numbers Pay special attention to the definition of algebraic numbers, and think about it. Understand these two facts: (1) Every rational number is algebraic. (2) Not every algebraic number is rational. One my show that not every real number is algebraic. One way to do it is to show that the set of all algebraic numbers is countable, but the set of all real numbers is not. Another way to do it is to find one specific real number and prove it's not algebraic; that requires Liouville's Theorem, which depends on some calculus. By the way, neither pi nor e is algebraic.

Exercises to turn in (no later than Tuesday 1/29): You may write up and turn in any one of the even-numbered Exercises in Section 1.1; and/or any one of the Exercises in Section 1.2. Notice that the back of the book provides help on some of the Exercises. ;
 

Read Sections 1.4 (the completeness axiom) and 1.5 (the infinity symbols).

Exercises to turn in (no later than Tueday 2/4): You may turn in for credit up to four of the 16 Exercises in Section 4.1 (pages 25-26). In addition, I'm going to state a proof-by-induction problem in class.

In lecture, I'm planning to do (1) an exercise involving the definition of supremum and infimum and (2) a proof that the rationals are dense in the reals. I'll also develop two more substantial and interesting results, one of which will deal with the geometric and arithmetic means.
 

The test on February 19 will have two parts. The first part will consist of short questions: True or False, give an example of something, state the definition of something, and such. The second part will give you three results and ask you to prove two of them.
 

The test on Tuesday the 19th may cover anything in the text up through Section 12 (but omitting Section 6), and may cover anything in lecture up through February 14.

I hope everyone will take the test at the scheduled time, but if for any reason you must miss the test, then please let me know about the problem as soon as possible.
 

The next topic is infinite series. We'll skip Section 13 for the moment. You should read Sections 14 and 15 carefully. On your own (not to turn in), do the odd-numbered problems in those sections.
 

On Tuesday, March 3, you may turn in for credit any or all of the seven even-numbered problems at the end of Section 14.
 

This week I plan to define a metric space and begin to discuss continuous functions in the setting of metric spaces. Read Section 13.

On Tuesday, March 12 you may turn in for credit any or all of the four even-numbered problems at the end of Section 15.

On Thursday, March 14 you may turn in for credit any or all of these problems in Section 13: 10, 12, 14 (the number defined in this problem is called the diameter of the set),
 

Read again Section 13. Do the odd-numbered problems in that Section on your own, not to turn in. Read pages 156-158 and 160 in Section 21. You may turn in for credit on the day of the test any of the problems 2, 4, 6 in Section 21 (p. 163).

In the lectures this week, I plan to define a compact metric space and establish several equivalent conditions. Please read Ross's treatment of the Bolzano-Weierstrass and Heine-Borel Theorems and know what they say. I'm offering you some rather sophisticated mathematics this week, and it's important that you attend class, be alert, ask questions, take good notes. A part of Test 2 may come from this material, but any such question will be a small piece of it; I'll point out examples of what I mean.

Test 2 on Tuesday, March 19, an 80-minute test, will cover everything since the last test, sequences and series through Section 15, and the metric space material done in lecture. Half of the test will be definitions, examples, and true-false questions, and half will be proofs.
 

Read Section 17 and do at least half of the odd-numbered problems in that Section. Read also these notes on compactness.

To turn in for credit on Tuesday, March 26, do up to three of the even-numbered problems in Section 17.

On Thursday the 21st I will prove the equivalence of two ways to define continuity at a point. One is in terms of sequences, and the other is the delta-epsilon definition. My proof will be in the setting of arbitrary metric spaces. Compare page 116 in Ross. Of course, continuity on a set means continuity at each point of the set; and there's an equivalent definition of continuity-on-a-set in terms of open sets, as I showed earlier. Compare pages 160-162 in Ross.

I will prove that a compact subset of a metric space must be both closed and bounded. In Euclidean space (though not in every metric space), the converse is also true; see the Heine-Borel Theorem, p. 86.

I will prove, using the open-set formulations of the definitions of compactness and of continuity, the following two results. The composition of two continuous functions is continuous; compare Ross's page 122. The image of a compact metric space under a continuous function is also compact; to put it briefly: Continuity preserves compactness. It follows that a continuous function on a compact interval of the line attains its max and its min. See Ross's discussions on page 126 and pages 160-162.
 

Read Section 18. On your own, do at least half of the odd-numbered problems.

To turn in for credit on Thursday, March 28, do any one or two of these: 18.4, 18.6, 18.8, 18.10.

The lectures will continue on the material of Chapter 3, most of which is special to the real line. Professor Dorroh will substitute for me on Tuesday, since I have jury duty. On Thursday I'll talk about the Intermediate Value Theorem and about uniform continuity.

You'll find an Exercise that you can turn in on Tuesday, April 9.
 

Read Sections 19 and 20. On your own, do at least half of the odd-numbered problems in each Section.

To turn in for credit on Thursday, April 11, you may do up to five of the even-numbered problems in Section 19, and up to three in Section 20.

In lecture on April 9 I defined pointwise convergence and uniform convergence for a sequence of functions. I also showed that "uniform convergence preserves continuity" and gave you most of the proof that the metric space of bounded continuous functions from a metric space into a complete metric space, with the "sup" norm, is complete. You will find these results in the book as Theorems 24.3 and 25.4.

On Thursday I gave you an example of a sequence of continuous functions converging pointwise to a function that's not continuous. I proved again that the metric space C of continuous real-valued functions on [0,1], with the "sup-norm" providing the metric, is complete. I also showed that C contains a set that's closed and bounded but not compact (fit this in with the Heine-Borel and Bolzano-Weierstrass Theorems). I did so by producing a bounded sequence of functions in C that has no convergent subsequence. I also summarized the basic results of power series.
 

Read Sections 23, 24, and 25. On your own, do at least half of the odd-numbered problems in each of those Sections.

To turn in for credit on Thursday, April 18, you may do as many of the even-numbered problems in Sections 23, 24, and 25 as you wish.

This week I'll begin talking about differentiation, and about the material of Sections 28 and 29. On Tuesday I plan to define differentiability and prove Rolle's Theorem and the Mean Value Theorem. On Thursday I plan to prove the Chain Rule and derive the properties of the exponential function from its differential equation. There are other results in Sections 28 and 29 which I'll rely on you to read, like for example the formulas for the derivatives of a product and of a quotient (Theorem 28.3, p. 208).

Test 3 on Monday April 23, an 80-minute test, will cover the material outlined above for Weeks 9, 10, 11, and 12 (everything starting with March 21). Half of the test will be "small questions:" True-False, give examples, state definitions, find the radius of convergence, and such. In the other half, you will have a choice of proofs to give, and they will be proofs that I've discussed in class.
 

Thursday: I'll begin discussing the Riemann integral, with a view to covering most of the material in Sections 32, 33, and 34.

On your own, do at least five of the odd-numbered problems in Section 28. On Thursday, you may turn in for credit any number of the even-numbered problems in Section 28.
 

In the lectures: The Fundamental Theorem of Calculus and its applications.

On your own, do at least five of the odd-numbered problems in Section 29.
 

Thursday May 9 is the last day you can turn in exercises for credit. You may turn in any number of the even-numbered exercises in Sections 29, 32, 33, and 34 by May 9.

In the lectures: The basic change-of-variable theorem for integrals. A power series and its derived series have the same radius of convergence; the derived series converges to the derivative. A selection of results on metric spaces, concerning the distance from a point to a set, the distance between sets, when such a distance is attained.

On your own, do at least five of the odd-numbered problems in each of the Sections 32 and 33.

On your own, do at least five of the odd-numbered problems in Section 34.
 

The exam will be Friday, May 17, 5:30 - 7:30 p.m., in the usual classroom. It will be comprehensive, and it will have very much the same design as the three tests.