Math 4031-1, Spring 2024: Information for Students

Spring 2024

General Information for Math 4031--Section 1.


2:30 -- 3:20 PM, M W F.

Our class will meet in Room 136 Lockett. Our class begins on Wednesday, January 17, 2024, and our last class will be on Friday, May 3, 2024. Our final exam will be Fri., May 10 5:30pm - 7:30pm.

Leonard Richardson

Office 386 Lockett

Office Hours

  1. MWF 11:30 AM -- 2:30 PM in person in my office, 386 Lockett Hall. Please use these in-person office hours provided you feel healthy. If you feel ill, please use the following online office hours:
  2. TTh online only, by appointment, at this Zoom link: Just email in advance letting me know what hours this afternoon you are available and I will reply with a time to meet. There is a Zoom Waiting Room in case more than one person comes at the same time. However, I can meet with several of you at the same time if the students are comfortable with this.
I am available at many other times. Email first to make sure I'm able to meet with you. I answer email many times daily---usually quickly.



E-Mail Email is the quickest way to reach me.


Richardson, L., Advanced Calculus: An Introduction to Linear Analysis, John Wiley & Sons, 2008. ISBN 978-0-470-23288-0. There is a list of errata. If you find an error not on this list, please tell me.
The text is also available as a free e-book through the LSU Library at this link. Please download the free e-book as the complete pdf file, so you will have the full text on your computer or on an external drive. Last year some students depended on using the book online but couldn't do their homework because sometimes the library's link to the text was not working. So download the full text as one pdf file. The LSU bookstore sells the hard copy, in case you would like the actual book for your personal library, as do I. But don't let them sell you an e-book since this is available to you free of charge using the link given here.

Graduate Assistant

You will turn in homework in class on the due-date. If you must be absent on the due-date, you may email the scan files of your homework to the grader directly by class time: Mr. Gyaneshwar "G" Agrahari at , who will grade those homework problems that are to be turned in---the ones that are assigned in red boldface in the table below.
The grader will be available to answer questions about the homework grading at 141 Lockett Hall from 12:30-1:30 on Friday.
Please be sure to write your solutions neatly and carefully so that they can be read.

Please Note: If you are turning an assignment in by email, the best way to submit an assignment by email is with a device such as a tablet or a drawing board that enables you to write on the computer screen and save or convert to pdf. If you have no such device, you can use a scanner or a phone to photograph your work as jpg images. Then place the images, photographed in the correct order, on your computer screen. Highlight the whole group of pages and select print and then select print or save to pdf. That should make one pdf file with all your pages in order. Thank you.
Homework and Tests All homework and tests will be submitted on paper in class, except as noted above. However, if you must be absent on a day when homework is collected, you may submit your assignment by class-time directly by email to the grader in pdf format as a single pdf file. (The homework will be submitted to the graduate assistant grader, not to me, your teacher.) If you have a tablet device you may compose your homework directly as a pdf file. Or, if you have a scanner/printer that will copy your handwritten work directly to a single pdf file, that is fine. Otherwise, you can use your phone to photograph each page, in the correct order, in jpeg. Place all the jpeg files on your desktop, highlight the group, right click the mouse on the group, select PRINT, and select PRINT to PDF. That should do the job, giving you a single pdf file with all the pages in the correct order. Be very sure to check your file for legibility before submitting. Make sure your writing instrument is dark enough and your light suitable for a clear, readable pdf file. We cannot grade what we cannot read. Thank you for being very careful and considerate about this. Your graded homework and graded tests will be returned to you with corrections as soon as possible. Save your graded work!

Math Major Requirements and Recommendations

Math 4031, followed by either Math 4032 or Math 4035, satisfies the Advanced Calculus requirement for the Mathematics major with a mathematics concentration. It prepares students for graduate study of mathematics and its applications. The Department strongly recommends that Mathematics majors planning graduate study in Mathematics take all three Advanced Calculus courses: Math 4031, 4032, and 4035.


Either MATH 2057 or 2058, and 2085 or 2090, or equivalents.


Attendance is required and you will be responsible for classwork on examinations. Your presence and participation in class is an essential part of this course. Do not miss class without a valid excuse. When you are absent, you are missed. If you must miss class, please keep track of where we are in the syllabus online, and be sure to visit my daily office hours so that I can help you to keep up with the work you missed.
Statistics show a strongly postive correlation between regular attendance and test grades. Most LSU students are conscientious and sensible about coming to class unless there is a serious, excusable reason for not being able to do so. However, there is an unfortunate tendency for some students to become discouraged as the term progresses and to cease regular attendance. This happens despite the fact that a student who is feeling discouraged has an especially great need to be in class and to ask questions. I encourage every student to come to class even when the going gets tough.
I will require documentation always if you are absent from an hour test or from the Final Examination.

If you are unavoidably absent on a day when homework is due to be turned in for grading, email a pdf scan file or a clear photographic image of your homework solutions directly to the grader before class time.

Homework is required and will be part of your final grade

Problems, mainly proofs, will be assigned frequently: approximately 3 assignments every two weeks. The assignments are your main work in this course. You are encouraged to seek hints to help you get started with these problems! It is required to turn in every assignment! The key to learning to prove theorems lies in how you study Advanced Calculus. It is very important to understand thoroughly how and why the proofs presented in the book and in class work. Please read the Introduction to your textbook! We will go over every collected homework problem in class, to help you prepare for tests. At the end of the course, your homework average on a 10-point scale will be added to your Exam average to produce your final average. For example, if your average on the homework is 5 points out of 10, and you have an 85% exam average, your final average would be 90%. In this example the homework credit would raise your grade from B to A-. This is an increase of two grade levels on LSU's +/- grading system. Proofs assigned for homework are a very important learning experience. Some students try a shortcut - copying the correct proofs from the board after the homework has been graded, without turning in their own efforts. This tends to produce proofs on tests that are written by rote from memory, and these tend to be lacking in logic and thus incoherent. It results also in low grades on Part I of each test, because the student's own conceptual errors have not been turned in and thus have not been corrected. Remember that homework is required! In order to learn the logical structure of advanced calculus, one needs to follow a given set of definitions and theorems from start to finish. If you wish to use other definitions or theorems from a different book, you must also include a proof that the definition or theorem you have chosen is equivalent to the one we used in the course. This will require that you do much more work than is needed to follow the definitions you have been given in our course. There are unscrupulous businesses online that will sell you solutions to homework problems. If you were to avail yourself of such a service, then at best you would be cheating yourself out of this part of your education. The result will be an unacceptably low grade and very likely the need to repeat the course and pay tuition a second time for the same course. Moreover, I have seen some of these illegal and unauthorized solutions to problems in my book for sale online that were utterly wrong and must have been written by someone incompetent in mathematics. Buyer beware!! Your learning of Advanced Calculus will come only from your own work. There are no shortcuts. You need to turn in every assignment on time, come to class daily from the first day of the semester to the last, ask questions about everything you do not understand clearly, and ask questions about any errors indicated on your returned homework assignments.

When should you ask questions?

You should ask questions every time you do not understand something and also every time you are curious about something. Ask questions in class. Be aware that when I am writing on the whiteboard I have no way of seeing your raised hand. So speak up with your questions! Our class size is small so you should view our meetings as a two-way discussion and not a formal lecture. It is a good thing to speak up with your questions!
Ask questions after class. Ask questions in my office hours. Ask questions by email. Please consider this: I have been teaching as a University faculty member for more than half a century. So I ask you to consider that I just may be able to help you with whatever is causing you difficulty if you will permit me to do so. It is a pleasure for me to help each and every student, so please give me a chance to help you to the best of my knowledge and ability. Every single one of you is important to me.

Lateness and Classroom Conduct

Please try to arrive on time for class. But sometimes it may be unavoidable to be late. If you are late, please come right into the classroom, doing so as quietly as you are able so as not to disturb other students. You should have as much class time as possible, so please just come in--quietly--and join the class even if you are late. Also, if homework is due that day, remember to turn it in on paper to me. Class time is a time for work. So when class begins please turn your attention to the work of the class.


We plan to have 3 hour tests, and they will be closed-book tests. No notes, whether on paper or electronic, are allowed. No communication devices are allowed. Part I of each hour test will consist of a choice of 8 out of 12 short answer questions, and Part II will offer a choice of 2 out of 3 proofs. (The Final Exam will be equivalent to two hour tests.) The proofs will be modeled closely on the collected homework, and they are sometimes identical. The short questions will be small variations of homework problems---including those not collected---together with examples from the lectures and notes. Thus if you have been coming to every class and have done the homework conscientiously, you should be prepared well for all tests. If you must miss a test, it is your responsibility to speak to me as soon as possible to determine whether or not your excuse is acceptable.


We plan to have three hour tests, worth 100 points each, and a two hour final examination, worth 200 points. Your test average TA will be the sum of your final exam grade and your hour test grades divided the maximum possible cumulative score, expressed as a percentage. Let HA denote your homework average on a 10 point scale. Your Final Average FA will be FA = TA + HA . (Alternatively, if it will benefit you, instead of adding your HA to your TA, we will replace your lowest test grade with your HA converted to a 100% scale. But experience shows most students benefit most from the calculation with TA + HA.) Thus 0<= FA <= 110. The minimum grade for each letter grade is as follows:
A+, 97
A, 93
A-, 90
B, 83
B-, 80
C+, 77
C, 73
C-, 70
D+, 67
D, 63
D-, 60
F, below 60
You should save all your graded work for future study and in case you think your final grade is in error. Please take note: Number of Dropped Grades: 0 Curve: None.

General Advice

  • Many students need help to learn how to write proofs. If you feel confused, it is important to see me for help as soon as possible. If you don't know how to start a homework problem, ask for a hint---either in class or in my office---or even by email. If you ask me a question about the homework, or if you email such a question to me, I may be able to think of a good hint and then I would email it to the whole class as a hint. I can guarantee you it is possible to learn to write sound proofs: Learning begins with your efforts and your persistence. If you are feeling 'lost' please see me privately in my Zoom office hour location so that I can help you. I am very happy to meet with each individual student to help.

  • "Why is it important for me to be able to write proofs?" Read an opinion by a distinguished expert at this link!

  • Attend class and ask questions. Non-attendance or lax attendance is usually the first sign of impending academic difficulty. Sometimes a student feels frustrated because of not understanding the classwork. If that is the case, it is necessary to ask more questions. Advanced calculus is a subject you can learn---but you must participate in this work.

  • Assignments to be turned in by the start time of class. Do not turn it in later than that, because it is not fair to the graduate teaching assistant, who will be busy enough with the work of grading the assignments that are turned in by the proper time.

  • LSU offers extensive academic support services to help students adjust to the demands of university studies: List of Frequently Used Services. HOWEVER, there are businesses both locally and online that sell their own instructional services to students for fees. LSU does not authorize or approve any such services and does not recognize their competence to serve your educational interests. Advertising for any such commerical service is prohibited in our classroom. If you see such advertising being circulated in our classroom please report it to me. And buyer beware!

  • Caution about Online Sources: Many of us make use of online information occasionally. But students need to be aware that your homework must be logically consistent with the development of Advanced Calculus in our course. What is an exercise in one course may be a theorem in another course. And the order of logical development varies from one course to another. Also, remember that online sources, such as UTube, are not universities. They have no faculty and no academic accreditation. Also, proofs generated by artificial intelligence are apt to be completely lacking in logic. They are just chains of words that commonly follow one another somewhere in a data base. When it comes to learning advanced calculus, there are no shortcuts. We need to do our own work to cultivate a strong sense of logical reasoning.

Homework Assignments and Downloads

We will update the list of assignments and tests below as the semester progresses. You will know an assignment has been updated if a due-date appropriate to this semester appears in the left-hand column. However, sometimes we will assign a problem for a certain date and then postpone it because we don't cover as much as planned in class. So check regularly for updates as to what is due and when. If you email me about a pending assignment, I may send a hint to the whole class in answer to your question, not giving your name of course!

Academic Honesty

The University has clear policies requiring academic honesty. If you get an idea from another book or an online source, or from talking with a friend, academic honesty requires that you acknowledge your sources openly. Above all, never copy directly from another person's written work as though it were your own. Remember that your own good name is irreplaceable. This is a sound principle which will serve you well throughout your life. Moreover, on a practical level, it is very foolish claim as your own an argument from a former student in this class or from a textbook. The arguments which are copied can be recognized very easily as not coming from the student, and often the precise source can be identified readily. This means that the honorable course of action is also the practical one.

Due Date

Assignments: Hand in problems in red bold face for grading. The problems in red bold face are required. Assignments must be written neatly so that the grader can read them. There is also a class of optional problems, called Bonus Problems, which are intended for those students who find the required homework easy and want to be seriously challenged. These are worth up to 20 extra homework points per problem. Bonus problems need to be emailed in pdf format directly to me on a separate sheet from the regular homework, clearly marked Bonus Problems at the top. Bonus problems are due, if you choose to do one of them, one full week after the date listed, unlike normal graded homework, which is due the date listed. Bonus problems must be handed in separately from the normal homework, and they will be graded more strictly for logical rigor than the required homework. Please read the Academic Honesty policy above!

January 17

Read this syllabus and bring any questions you have to class today.

January 19

Read Pp. xxi--5.

January 22

Prove that the vector product (also called the cross product) of vectors in 3 dimensional space, does not satisfy an associative law for multiplication. (Hint: find a simple example of a cross product of 3 vectors that fails to be associative, showing the calculations. It is easiest to do this with the standard unit vectors: i, j, and k. )

Do problems 1.1, 1.4--1.6, 1.8, 1.10, 1.11. These are not to hand in, but you should write your solutions on paper in order to learn from the work. We will go over some---but not all---of these problems in class according to your requests. Let's be partners in this work: You should ask me about the questions you would like me to solve on the board. You will be responsible for all assigned problems, whether collected or not.

January 24

Hand in: 1.3, 1.7. Please remember to write neatly so the grader can read your work, and put your name on your paper so he can record the grades!

(Remember: Only problems in RED BOLDFACE are to be handed in by class time to me for grading!)

January 26

Hand in:
  1. Use the result of homework problem 1.3 to prove that the real number 1 is positive. That is, prove that 1>0. (Hint: Use the Axiom of Trichotomy to see that there are only 3 potential cases for the number 1. Show that two of those cases lead to an impossibility, leaving only one possibility.)
  2. Do problem 1.9.
Please be sure to write your solutions neatly, so that the grader can read them!

January 29

Read from Example 1.3 through Definition 1.2.4. Then do: 1.12 -- 1.16, 1.18, 1.20--1.21, 1.23 -- 1.24, 1.26, 1.27.

January 31

1.19, 1.22. Downloadable Optional Bonus Problem B1 to hand in a week from today. Bonus Problems are intended for those students who find the required homework easy and want to be seriously challenged. These are worth up to 20 extra homework points per problem. Bonus problems need to be emailed directly to me, your teacher on a separate pdf file from the regular homework, clearly marked Bonus Problems at the top. Bonus problems are due, if you choose to do one of them, one full week after the date listed, unlike normal graded homework, which is due the date listed. Bonus problems must be submitted by email to me, at, separately from the normal homework, and they will be graded more strictly for logical rigor than the required homework.

February 2

1.31, 1.33.

February 5

1.28, 1.32. (Hint for 1.32: Remember that the supremum and the infimum of a set might or might not be in the set. So don't make any unwarranted assumptions!) Also: 1.29, 1.30, 1.34--1.39.

February 7

1.41 --1.44.

February 9

1.47--1.50, 1.53--1.55.

February 14

1.51, 1.57. Also: 1.59--1.61, 1.63. (Hint for 1.51: Do not compute with (or even write) lim xn or lim yn until you have proven that these limits exist. Otherwise, you will be operating with undefined terms.) (Hints for 1.57: It may help to introduce the notations sn(x) for the supremum of the nth tail Tn(x) of the x sequence, and similarly for the y sequence and the x+y sequence.) Please note that the Bolzano-Weierstrass theorem, the Nested Intervals theorem, and the Heine-Borel theorem will require your focused attention! Here is an optional bonus problem B2, 1.56, for those students who are looking for more challenging problems than those in the required homework. Email this to me,, separately from the regular homework one week from the due date printed to the left of this box, if you choose to do it. Bonus problems will be graded more strictly for logical rigor than the required homework. Remember, bonus problems will be be emailed to me, and not handed in with the regular homework.

February 16

1.62, 1.64; Also 1.67-1.70. Please note that the Bolzano-Weierstrass theorem, the Nested Intervals theorem, and the Heine-Borel theorem will require your focused attention!

February 19

1.71, 1.72. Here is another optional bonus problem B3, for one week from the date to the left if you choose to do it. This question is similar to 1.70, but not identical! Suppose an is a strictly increasing sequence, bn is a strictly decreasing sequence, and an < bn for all n. Prove or Give a Counterexample: The intersection from 1 to infinity of (an,bn) is nonempty.

February 21

1.76--1.80, 1.82, 1.85, 1.86. Be careful with terminology! If E is contained in the union Ua in AOa of open sets Oa then it is the set {Oa | a is in A} that is an open cover of E. Do not confuse the open cover with its own union. The union of a family of sets is only one set!

February 23

1.83, 1.84. Be careful with terminology! If E is contained in the union Ua in AOa of open sets Oa then it is the set {Oa | a is in A} that is an open cover of E. Do not confuse the open cover with its own union. The union of a family of sets is only one set!

February 26

Bring questions to review for Hour Test #1! Everyone is requested to make a list of questions. Your questions will help to make the review more useful for yourself and for the whole class. A previous first hour test is available below for you to download. Note however that what we cover in the course does vary somewhat from year to year, as does the coverage of the tests. Our test will cover all assignments due by today.

February 28

First Hour Test. This test will cover all assigned work that was due before today.

February 29

Please download and read carefully the Hour Test #1, Spring 2024, Solution Sketches and Class Statistics.

March 4

1.87, 1.89, 1.90, 1.92, 1.94.

March 6

1.88, 1.91. Here are two more Bonus Problems, for those who seek more challenging exercises: B4 - 1.93, and B5-1.96. If you choose to do these, turn them in separately from the regular homework a week from today. After any bonus problems are graded and returned to you, please feel free to come to my office to ask for correct solutions.

March 8

2.1--2.5, 2.13--2.15. Here is another optional bonus problem, for a week from today if you choose to do it: B6-Problem 2.16.

March 18

2.6, 2.7 , 2.8; Be sure to read Cor. 2.1.1 and its proof to see how the sequential criterion for limits of functions is used. Read also Definition 2.1.3. (Hints: For 2.6, use the sequential criterion to prove the limit of the function does not exist. Do not use L'Hopital's Rule for 2.7(b)! Instead, draw a unit circle and find useful inequalities by comparing areas of triangles and a circular sector, expressible in terms of x, sin x, and tan x.)

March 20

2.19--2.20, 2.22-2.23, 2.25.

March 22

2.21, 2.24, 2.27. For 2.27, follow the sequence of steps given. This exercise is a theorem discovered by Cauchy. For optional bonus credit: B7-2.26 and B8-2.28. These two would be for one week from this date.

March 25

2.29--2.34, 2.40--2.41, 2.43--2.47.

March 27

Bring Questions to Review for the Second Hour Test today! This test will cover the assignments that were due after the first hour test.

March 29

Good Friday Holiday

April 1

Second Hour Test today!

April 2

Please download a copy of the Spring 2024 second hour test from this link: Hour Test #2, Spring 2024, with solution sketches and overall class statistics at the end..

April 3

2.35, 2.37, 2.42. In problem 2.42, remember to prove your conclusions for each of the three stated questions!

April 5

2.48, 2.50, 2.52-- 2.59.

April 8

2.49, 2.51. For 2.49, see Example 2.6.

April 10

LSU Campus Closed Today for Weather Emergency! Please use this link to the Zoom lesson to make up for today's missed class. 2.60 --2.63, 2.65 -- 2.67, 2.69. (2.70 is optional since we don't have time for Dini's theorem in class this semester. You are welcome to read it and ask me if you have questions about it. Dini's theorem is an interesting partial converse to the theorem that uniform convergence preserves continuity of functions.)

April 12

2.64, 2.68 . In these exercises, you may use derivatives and L'Hospital's rule although they do not appear in this text until later.

April 17

3.1, 3.2, 3.4--3.9, 3.11, 3.12. These problems that are not to hand in but are important. Here is an optional bonus problem that would be due 1 week from today: B9- 3.14.

April 19

3.3, 3.10 For problem 3.3 use only the definition of the Riemann integral to show for each epsilon >0 there exists a delta >0 such that ||P||< delta implies that |P(f,{x_i bar}) - 0|< epsilon. For 3.10 you will benefit from the hint in the statement of the problem in the text.

April 24

3.18--3.23. Please read the statement of Theorem 3.24 and use it freely in the homework. It is a very powerful theorem.

April 26

3.24 .

Bring questions to review for the third hour test!

April 29

Third Hour Test today! This test will cover the work that was due since the second hour test.

April 30

Please download a copy of the Spring 2024 third hour test from this link: Hour Test #3,Spring 2024, with solution sketches and overall class statistics at the end. and study the solutions.
May 1

3.25, 3.26, 3.30b. ((Hint for 3.25: In 3.25, the strictness of the inequality is the whole problem.) (Hint for 3.30b: Only part (b) is assigned, but here is a hint anyway for the unassigned part (a): In 3.30(a) one of the directions of implication has already been proven in the text. The other direction remains to be proven. (Hint: Use theorem 3.2.4, the variant of the Darboux Integrability Criterion.) Also: 3.27.

May 3

3.34 -- 3.37. Bring questions to review for the Final Exam! Also, please remember to fill out the end-of-course evaluation form that is available to you online! Your anonymous feedback is very important: It helps your teacher and the University to serve your needs to the best of our ability. Remember: Your opinion matters: You are the reason we are here! Thank you..
Don't forget to review from the beginning
of the course! This 200-point exam will cover the whole course in a uniform manner, so remember to review from the beginning of the course. Your final grade for the course will be the larger of the following two: 1. The grade guaranteed by the formula provided higher on this page. 2. One letter below the final exam grade. Thus the final exam provides a safety net that supplements the calculations specified above. As an extreme example (that has not actually occurred in my experience) one might have a failing final average and an A on the final exam, resulting in a B for the course. But less extreme examples do in fact occur, showing the value of catching up by the end of the course.

Exam Week Exam week Office Hours: Monday & Wednesday 1PM -- 3PM, Friday 3PM--5PM, in 386 Lockett. Email me as ususal if you have questions and can't make it to my office hours. I can arrange a Zoom office hour for Tuesday late afternoon by request.
May 10 Our final exam will be Fri., May 10, 5:30pm - 7:30pm, in our usual classroom.

Please download Final Exam, Spring 2024, Solution Sketches and Class Statistics.