- May 11
- I posted grades at the TAMU
eLearning site. On the final exam, the median was 88, the
mean was 87, and there were two scores of 100. Great
job!

For the course averages, the median was 86 and the mean was 85. - May 7
- The final exam was given, and solutions are available.
- April 30
- I have posted the homework averages (based on the highest ten homework scores) at the TAMU eLearning site. For the class as a whole, the median homework average is 86 and the mean is 82.
- April 29
- At our last class meeting, we reviewed for the final examination, did the hard-copy course evaluations, and worked on the true/false exercises in sections 4.4, 4.5, 5.1, 5.2, 5.3, and 5.4.
- April 27
- We discussed improper Riemann integration.

Thursday will be our last class meeting for the semester. - April 22
- We discussed the fundamental theorem of calculus.

I posted at the TAMU eLearning site solutions to both assignments due this week.

The assignment for next time (not to hand in) is to compile a list of the major named theorems from the course. - April 20
- We discussed the integrability of monotonic functions and continuous functions as well as Lebesgue's characterization of Riemann-integrable functions as functions whose discontinuities form a set of measure zero.

The assignment for next time is Exercise 5.1.4 on page 139 and Exercise 5.2.10 on page 152. - April 19
- I made some minor updates to the comments on Chapter 5.
- April 15
- We discussed Riemann's original approach to integration (via “Riemann sums”) and some properties of the class of integrable functions.

The assignment that was originally due today is held over until next time, with the addition of Exercise 5.2.3a on page 150. - April 13
- We discussed Darboux's approach to the Riemann integral through upper sums and lower sums.

I posted at the TAMU eLearning site solutions to the assignment due today.

The assignment for next time is Exercises 5.1.1a and 5.1.3 on pages 138–139. - April 8
- We discussed inverse functions and their derivatives.

I posted at the TAMU eLearning site solutions to the assignment due today.

The assignment for next time is Exercises 4.5.1, 4.5.2, and 4.5.9 on pages 128–129. - April 6
- We discussed two applications of the mean-value theorem—l'Hôpital's rule and Taylor's formula—and looked at some examples of using these theorems.

The assignment for next time is Exercises 4.4.4 and 4.4.5f on page 124. - April 5
- I finished grading the second exam, which turned out to be more challenging than the first exam. The person who had the high score of 99 on the first exam made a raw score of 90 on the second exam, so I renormalized the scores on the second exam by adding 9 points. The adjusted scores, which are posted at the TAMU eLearning site, range from a high of 105 to a low of 61, with a mean of 84 and a median of 83. I will return the graded exams in class tomorrow (April 6).
- April 1
- The second exam was given, and solutions are available.
- March 30
- We reviewed for the exam to be given on Thursday and worked in groups on the true/false exercises from sections 4.1, 4.2, and 4.3.

I posted at the TAMU eLearning site solutions to the assignment that was due today. - March 26
- I updated the comments on Chapter 4.
- March 25
- We discussed the statement and the proof of the mean-value theorem along with some applications.

I posted at the TAMU eLearning site solutions to the assignment that was due today.

The assignment for next time is Exercises 4.3.1d, 4.3.3, and 4.3.8 on page 116.

Reminder: The second exam takes place next week on Thursday, April 1, covering Section 2.4, Chapter 3, and the first three sections of Chapter 4. - March 23
- We discussed three different formulations of differentiability, looked at some examples, and considered some of the properties of derivatives.

The assignment for next time is Exercise 4.1.1b on page 104 (note that a≠0 should say a>0) and Exercise 4.2.2 on page 107.

Reminder: The second exam takes place next week on Thursday, April 1, covering Section 2.4, Chapter 3, and the first three sections of Chapter 4. - March 21
- I updated the comments on Chapter 4.
- March 11
- We worked in groups on the true/false exercises in Chapter 3.

I posted at the TAMU eLearning site solutions to the assignment that was due today.

The assignment for Spring Break is to have fun and to be safe. - March 9
- I posted at the TAMU
eLearning
site solutions to the assignment that was due today.

In class, we discussed the notion of uniform continuity, looked at some examples, and saw the theorems that (i) a function with a bounded derivative on an interval is necessarily uniformly continuous on that interval; (ii) a continuous function on a closed, bounded interval is automatically uniformly continuous; and (iii) a function on a bounded subset of the real numbers is uniformly continuous on the set if and only if the function preserves Cauchy sequences.

The assignment for next time is Exercises 3.4.1a and 3.4.6 on page 96. - March 5
- I posted at the TAMU eLearning site solutions to the assignments that were due on March 2 and March 4.
- March 4
- We discussed the notion of continuity on a general subset of the real numbers (not necessarily an interval), and we looked at the question of which functions map Cauchy sequences to Cauchy sequences.

The assignment for next time is Exercise 3.2.6 on page 82 and Exercises 3.3.4 and 3.3.5 on page 91. - March 2
- We discussed continuity of real functions, some examples, and the statements of the extreme-value theorem and the intermediate-value theorem.

The assignment for next time is Exercises 3.3.1b, 3.3.2a, and 3.3.6 on pages 90–91. - March 1
- I finished grading the first examination. The supremum of the set of scores (equivalently, the maximum, since the set has finitely many elements) was 99, the infimum was 54, the mean (average) was 85, and the median (middle score) was 88. Good job!
- February 25
- We discussed two ways to define the limit of a function: either through the intervention of sequences, or instrinically.

I posted solutions to the assignment due today at the TAMU eLearning site.

The assignment for next time is Exercises 3.1.1c and 3.1.7 on pages 74–76 in Section 3.1. - February 23
- We discussed the notion of a Cauchy sequence, looked at some examples, and proved that a sequence of real numbers converges if and only if it is a Cauchy sequence.

The assignment for next time is Exercises 2.4.2 and 2.4.3a on page 61 in Section 2.4. - February 22
- I was out of town last week due to a death in the family. The first exam was given on February 18, and solutions are available.
- I also updated the comments on Chapter 2.
- February 14
- I posted solutions to the third and fourth homework assignments at the TAMU eLearning site.
- February 11
- I posted solutions to the second homework assignment at the
TAMU eLearning site.

In class, we proved some theorems about limits of sequences: a convergent sequence is necessarily bounded; a bounded monotonic sequence is convergent; every bounded sequence has a convergent subsequence (Bolzano–Weierstrass theorem).

The assignment for next time (not to hand in) is the true/false exercises 2.1.0, 2.2.0, and 2.3.0.

Reminder: The first examination is scheduled for Thursday, February 18. It covers Chapter 1 and Sections 2.1–2.3. - February 9
- I posted solutions to the first homework assignment at the
TAMU eLearning site.

In class, we used the definition of limit to prove some theorems about limits, and we looked at the definition of a limit of a sequence being infinite.

The assignment for next time has one part to hand in and another part not to hand in.- To hand in: revise and correct your solutions to the second homework.
- Not to hand in: Exercises 2.2.1a and 2.2.5 on pages 51–52 (Section 2.2).

- February 8
- I changed my Wednesday office hour for the rest of the semester to 13:00–14:00. My office hour on Tuesday and Thursday remains 15:00–16:00.
- February 5
- I updated the comments on Chapter 1 with a remark about Exercise 1.4.5. Also, I have posted at our class website inside the password-protected TAMU eLearning site the first chapter of
*A Primer of Real Functions*. This material is intended for the honors section, but all interested students are welcome to read it. - February 4
- We discussed the definition of the limit of a sequence and worked two examples in detail.

The assignment for next Tuesday (February 9) is Exercises 2.1.1c and 2.1.6 on pages 45–46 (Section 2.1).

The assignment for next Thursday (February 11) is to revise and correct your solutions to the second homework (which was returned in class today). - February 2
- We continued the discussion of cardinality of infinite sets. We saw that the following sets are countable: the integers, the Cartesian product of two countable sets, every infinite subset of a countable set, the rational numbers, the union of two countable sets, and more generally the union of a countable number of countable sets.

The assignment for next time, as previously announced, is to revise and correct your solution to the first homework assignment and resubmit it (unless you scored a 10 the first time). - January 28
- We discussed some of the true/false exercises in
Chapter 1, and we examined Cantor's diagonal argument for
proving that the set of natural numbers has smaller cardinality
than the set of real numbers.

The assignment for Tuesday (February 2) is Exercise 1.5.7 on page 35 (the boxed exercise) and Exercise 1.6.6 on page 40 (about the pigeonhole principle). Notice that part a of Exercise 1.6.6 is a special case of part b; the idea is that if you solve part a, then you should be able to generalize to get part b without much additional work.

The assignment for Thursday (February 4) is to revise and correct your solution to the first homework assignment and resubmit it (unless you scored a 10 the first time). - January 26
- We discussed some properties of functions (injective,
surjective, and bijective) and of inverse images (preservation of
unions and intersections). We worked in groups on some of the
true/false questions (1.2.0, 1.3.0, 1.4.0, and 1.5.0).

The assignment for next time (not to hand in) is to continue working on these true/false questions. - January 25
- I updated the comments on Chapter 1.
- January 21
- We discussed the order relation on the real numbers, the
completeness property of the real numbers, and the notions of
supremum and infimum.

The assignment for next time is Exercises 1.2.7a, 1.2.8a, and 1.2.9 on page 15 and Exercise 1.3.1f on page 22. - January 19
- We did some examples of proofs by induction (Section 1.4
in the textbook), and we started talking about the real numbers as
a complete ordered field. We began with the definition of a field
and with some examples of fields.

The assignment to hand in next time is boxed Exercise 1.2.3 on page 14 (about the positive part and the negative part of a real number) and boxed Exercise 1.4.5 on page 28 (proof of an inequality via induction).

Next time we will discuss Sections 1.2 and 1.3 and then continue on in the order of the sections in the textbook. - January 18
- I added comments on three of the appendices.
- January 15
- Welcome to Math 409. I will be regularly updating this page
with homework assignments, brief summaries of what we did in class,
and other information.

Reading the textbook is essential. If you encounter unclear points in the book, please let me know, and I will update my comments on the textbook.

A student asked for suggestions of additional books that could be used for supplementary reading. Here are some.*Introduction to Real Analysis*by William F. Trench, free pdf download.*Elementary Real Analysis*, second edition, by Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner, available both as a pdf download designed for onscreen viewing and as an inexpensive paperback.*A Primer of Real Functions*by Ralph P. Boas, fourth edition, revised and updated by Harold P. Boas, Mathematical Association of America, 1996, ISBN 978-0-88385-029-9.