This course ended May 11, 2008

Math 331: Foundations of Analysis I

   Department of Mathematics and Computer Science
   Hobart and William Smith Colleges

   Spring 2008.

   Instructor:  David J. Eck.

   Monday, Wednesday, Friday, 12:20 to 1:15.
   Room Eaton 105.
   Course handout:

Fourteenth Week: April 28 and 30; May 2

We will wrap up the course material this week by spending the entire week on Section 4.6, Series of Functions. The final homework assignment is due on Friday. Sometime this week, we will decide on the exact format of the final exam.

Thirteenth Week: April 21, 23, and 25

After covering alternating series and conditional and absolute convergence on Monday, we will spend the rest of the week on Section 4.5 which introduces sequences of functions, concentrating on uniform convergence and its consequences. The following homework on that section is the final homework assignment of the term and is due next Friday, May 2:

           Section 4.5: 3, 7, 8, 9, 12

Twelfth Week: April 14, 16, and 18

We will finish up infinite sequences (Section 4.2) on Monday. The main topic that we still have to cover is Cauchy sequences. We will then move on to infinite series. We will cover Section 4.3 and part of Section 4.4 before the end of the week. The following homework is due next Friday, April 25:

            Section 4.3:  3, 6, 7 
            Section 4.4:  3adf

Eleventh Week: April 9, 11, and 13

After the test on Monday, we begin Chapter 4, which covers sequences and series. We should finish through Section 4.2 (sequences) by Monday of next week. The following homework is due next Friday, April 18:

            Section 4.1:  3, 6, 10, 14
            Section 4.2:  3, 7, 11, 13

The take-home part of the test is due on Friday of this week.

Tenth Week: March 31; April 2 and 4

Remember that there is a test next Monday, April 7. The in-class part will cover Chapter 3, sections 1 to 6. The take-home part will cover those sections and might also have questions from Section 3.7.

We will finish Section 3.6 on Monday with a presentation of the Second Fundamental Theorem of Calculus. We will then begin Section 3.7, which we should complete by the end of the week. There is an assignment due on Friday.

Ninth Week: March 24, 26, and 28

We will continue with Chapter 3, finishing Section 3.4, covering Section 3.5, and startings 3.6. There will be student presentations on Wednesday, Friday, and next Monday of some of the major theorems in these sections. The following homework is due next Friday, April 4:

             Section 3.4:   # 2, 5
             Section 3.5:   # 4
             Section 3.6:   # 3, 8, 12

Eighth Week: March 10 and 12

We will continue with Chapter 3, and we will be starting in on integration by Wednesday. The following homework is due on Wednesday:

             Section 3.2,  # 4, 7
             Section 3.3,  # 2, 7, 10

There is no class on Friday because of Spring break. Classes resume on March 24. Have a great break!

Seventh Week: March 3, 5, and 7

We are at the beginning of the material at the heart of the Calculus: the derivative and the integral. The goal is to cover the basic material in a rigorous way, building on the foundation that was laid in Chapters 1 and 2. The reading for the week is Sections 3.1, 3.2, and 3.3. The following homework is due on Friday:

          Section 2.6,  # 12
          Section 3.1,  # 1a, 3b, 10, 11, and 12

Sixth Week: February 25, 27, and 29

After the test on Monday, we will finish up Chapter 2 and we will begin Chapter 3. Your main work for the week is to finish the take-home part of the test, which is due in class on Friday.

Fifth Week: February 18, 20, and 22

There is a test on Monday of next week. There will be no homework assignment due next week, but the take-home part of the test will be due in class on Friday. Here is an information sheet about the test:

Information Sheet for Test #1

In class this week, we will finish up Chapter 2. The main topics for the week are continuity and uniform continuity. The last section of Chapter 2 covers various properties of continuous functions on a closed interval, including the fact that any continuous function on a closed, bounded interval is in fact uniformly continuous.

Fourth Week: February 11, 13, and 15

We will continue our rigorous look at limits and the theory of limits this week. We will finish Section 2.2 and will cover 2.3 and 2.4. We might begin working on continuity, Section 2.5, by Friday.

The first Math department colloquium will be given on Wednesday at 4:30 in room Napier 201. Vince Cassano '91 and Kim Oaks '85 will talk about their experience working as actuaries. There will be refreshments at 4:00.

The following homework is due next Friday:

            Section 2.4:  # 3, 8c, 10, 11, 12a, 13c
            Section 2.5:  # 7, 9

(For exercise 2.6.5c, compare Example 2.6.1 and Exercise 2.6.2.)

Third Week: February 4, 6, and 8

After finishing up Chapter 1 on Monday, we move on to Chapter 2. For the next week, through next Monday, we will be concentrating on limits, using the rigorous "epsilon-delta" definition of the limit of a function at a number a. On Friday, Josh will present a proof of the Closed Nested Interval Theorem, and Christine will use that theorem to give an alternative proof of the Bolzano-Weierstrass Theorem.

The following homework is due next Friday, February 15:

            Section 2.1:   7a, 7c
            Section 2.2:   1a, 1c, 1g, 1h, 2, 9, 12
            Section 2.3:   1, 6, 10a

In addition, Anna And Lisa will each be doing a presentation of some sort on the 15-th.

Second Week: January 28 and 30 and February 1

The reading for the week is Sections 1.3 and 1.4. Section 1.3 lists a set of 15 axioms that together completely characterize the real numbers. The axioms state that the real numbers form a complete ordered field. While you are not required to memorize all the axioms, you should know the order axioms and their consequences, and you should of course be very familiar with the Least Upper Bound Property. Section 1.4 covers two important theorems that are consequences of the least upper bound property: The Heine-Borel Theorem and the Bolzano-Weirstrass theorem. You should understand the statements of these theorems and their proofs.

The following homework is due next Friday, February 8:

            Section 1.3:  2, 11, 12, and 13
            Section 1.4:  2, 5, 6, 12, and 15

In addition, Josh will do Problem 7 from Section 1.4 and will present the solution in class.

First Week: January 21, 23, and 25

You should read Sections 1.0 and 1.1 before class on Wednesday, and read Section 1.2 before class on Friday. These sections explain how the set of real numbers (including the irrational numbers) can be constructed from the set of rational numbers. Some particular numbers are shown to be irrational. A basic property of the real numbers, the Least Upper Bound Axiom, is introduced.

The following homework is due in class next Wednesday, January 30:

      Section 1.1, problems  5, 12, 14, 16, 22
      Section 1.2, problems  4, 6, 13, 17, 18

Homework will ordinarily be collected at the beginning of class on the day when it is due. Sample solutions will be handed out at that time. Homework will not be accepted late (though in a few rare and extraordinary circumstances, it might be excused). Please remember that to get full credit on homework, you must show your work, including explanations written out in full sentences in English when appropriate.