 Augustin Louis Cauchy, 1789-1857 |
 Karl Weierstrass, 1815-1897 |
 Bernhard Riemann, 1826-1866 |
Augustin Louis Cauchy, 1789-1857 |
Karl Weierstrass, 1815-1897 |
Bernhard Riemann, 1826-1866 |
COURSE NUMBER: MATH 4127/5127
TIME: 2:55-4:15 TR; PLACE: Gilbreath 104
INSTRUCTOR: Robert "Dr. Bob" Gardner; OFFICE: Room 308F of Gilbreath Hall
OFFICE HOURS: 4:15-4:45 TR and by appointment; PHONE: 439-6979 (Math Office 439-4349)
E-MAIL: gardnerr@etsu.edu
WEBPAGE: Dr. Bob's faculty webpage
(see my webpage for a copy of this course syllabus, copies of the classnotes in PDF, and updates for the course).
TEXT: James Kirkwood's An Introduction to Analysis, 2nd Edition. This is in print by both PWS Publishing Company (Boston, MA; 1995) and Waveland Press (Long Grove, IL; 2002). A third edition is out from the CRC Press (Boca Raton, FL; 2021).
SUPPLEMENTAL REFERENCES: In this course, we give a rigorous development of calculus, the topology of the real line, and Riemann integration. Several of the results which we see will be familiar from your freshman calculus classes (in fact, a calculus book will make good supplementary reading). I will occasionally assign problems and cover material not in the text (time permitting). I will rely on the following sources:
COURSE DESCRIPTION: The 2024-25 ETSU Undergraduate Catalog describes this course as: "Studies elements of point set topology, limits and continuity, differentiability, Taylor's theorem, approximation, Riemann integral." These topics are covered in Chapters 1, 2, 3, 4, 5, 6, and Section 8.1.
PREREQUISITE: The catalog states that the prerequisites are Calculus 3 (MATH 2110) and Mathematical Reasoning (MATH 3000).
GRADING: Homework (H) will be assigned and collected regularly. We will have a midterm (M) and a final (F). Your average will be computed as follows:
HOMEWORK: Homework will be assigned and collected at roughly one week intervals. The homework problems will be almost exclusively from the text book. YOU MUST SHOW ALL DETAILS ON THE HOMEWORK PROBLEMS!!! Justify every step and claim you make - this is how you convince me that you know what you are doing. You may find some answers online, but these rarely sufficiently justify all steps and are unacceptable as homework solutions. In addition, you will be given hints as to how to work the homework problems and proofs; these hints will be part of the instructions for the homework and must be incorporated in your solutions. Homework will usually be due on Saturdays through DropBox in D2L. You will need to create PDFs of your homework to electronically submit it.
ACADEMIC MISCONDUCT: If you have any questions about the assigned homework problems, then I will try to address them in class. If you need additional information, then let me know. We can work it out through e-mail, Zoom, or in-person meetings in my office. I expect that the work you turn in is your own and that you understand it. Some of the homework problems are fairly standard for this class, and you may find proofs online or proofs generated by AI. The online proofs or AI generated proofs may not be done with the notation, definitions, and specific methods which we are developing and, since they are not your work, they are not acceptable for this class. If I get homework from two (or more) of you that is virtually identical, then neither of you will get any credit on that assignment. If your homework is identical to one of your classmates, with the exception of using different symbols/variables and changing "hence" to "therefore," then we have a problem! I will not hesitate to charge you with academic misconduct under these conditions. To avoid this, do not copy homework and turn it in as your own!!! Once you start a test, you must stay in the room until you complete it or until the class time is over. If you have a documented medical need to leave class during the test then this will, of course, be honored (provided you give me the documentation before hand). I will provide all paper for the tests. You will only need a pencil and eraser. You will put your backpack at the front of the classroom during the test and your phone will be placed on the table at the front of the room (so please make sure you have the phone silenced). Cheating on a test will result in a grade of 0 on that test. This is an example of academic misconduct and I will have to act on this as spelled out on ETSU's Academic Integrity @ ETSU webpage (last accessed 7/26/2025). When such a charge is lodged, the dean of the College of Arts and Sciences is contacted. Repeated or flagrant academic misconduct violations can lead to suspension and/or expulsion from the university. When such a charge is lodged, the dean of the College of Arts and Sciences is contacted.
DESIRE2LEARN: I will not rely on the Desire2Learn ("D2L") website for the posting of notes and supplements; all of this material is freely available on my faculty webpage and does not require a login. I will use D2L to collect homework (in DropBox) and to post your grades, my homework solutions, and recordings of class lectures.
CLASS NOTES: We will use projected digital notes for the presentation of definitions, examples, and proofs of some theorems. Copies of the notes are online. It is recommended that you get printed copies of the notes before the material is covered in class. This will save you from writing down most notes in class and you can concentrate on listening and supplementing the notes and examples with comments which you find relevant. You should read the online notes to be covered in class before each class (we will not have class time to cover everything in the notes; they are very thorough). Try to understand the examples and the meanings of the definitions. Ideally, you should also read each relevant section of the book, paying particular attention to examples.
OUTLINE:
Our tentative outline is:
Chapter 1. The Real Number System: Section 1-1, Sets and Functions, is assumed as background material that you already know from Mathematical Reasoning (MATH 3000). Section 1-2, Properties of the Real Numbers as an Ordered Field, includes the topics: definition of field, orderings (a positive set and the Law of Trichotomy), examples of ordered fields, axiomatic development of ℕ and ℚ, rational exponents, absolute value. Section 1-3, The Completeness Axiom, includes the topics: least upper bound, greatest lower bound, definition of complete ordered field, Axiom of Completeness (Axiom 9), uniqueness of a complete ordered field, the Archimedean Principle, definition of same cardinality of sets, countable and uncountable sets, the Cantor diagonalization argument, power set, cardinal number, Cantor's Theorem.
Chapter 2. Sequences of Real Numbers: Section 2-1, Sequences of Real Numbers, includes the topics: limit of a
sequence, divergent sequence, bounded sequence, sums/products/quotients/multiples of sequences and their limits, Cauchy sequences.
Section 2-2, Subsequences, includes the topics: subsequential limit, convergent sequences and convergent subsequences, classification of subsequential limits.
Section 2-3, The Bolzano-Weierstrass Theorem, includes the topics: limit point of a set of real numbers, Bolzano-Weierstrass Theorem, a bounded sequence has a convergent subsequence, lim sup and lim inf of a sequence, sup( f ) and inf( f ) for function f, a sequence of real numbers is Cauchy if and only if it is convergent.
Chapter 3. Topology of the Real Numbers: The one section in the chapter includes the topics:
open set, closed set, intersections and unions of open/closed sets, open/closed relative to a set, interior point/boundary point/limit point/isolated point of a set, closure of a set, cover of a set, open cover of a set, finite cover of a set, compact set, Lindelof Property, Heine-Borel Theorem, a set of real numbers is open if and only if it is a countable disjoint union of open intervals (WOW!).
Chapter 4. Continuous Functions: Section 4-1, Limits and Continuity, includes the topics: the definition of limit of a function, epsilon-delta proofs of limits, The Sandwich Theorem, the definition of continuity at a point, linear combinations products and quotients and compositions of continuous functions, compact sets and continuous functions, Intermediate Value Theorem, uniform continuity.
Section 4-2, Monotone and Inverse Functions, includes the topics: the idea of infinity, definitions of infinite limits of a function, definitions of the limit of a function as x approaches infinity, one-sided limits and continuity, monotone functions, discontinuities.
Chapter 5. Differentiation: Section 5-1, The Derivative of a Function, includes the topics: derivative of a function at a point and its epsilon-delta definition, higher order derivatives, differentiable implies continuous, Linearity/Product Rule/Quotient Rule for derivatives, increasing/decreasing and extrema ideas.
Section 5-2, Some Mean Value Theorems, includes the topics: Rolle's Theorem, Mean Value of Theorem, Generalized Mean Value Theorem, L'Hopital's Rule, the derivative of a function and the derivative of its inverse.
Chapter 6. Integration:
Section 6-1, The Riemann Integral, includes the topics: sup and inf of sets and subsets, partition of an interval, refinement of a
partition, upper Riemann sum and Lower Riemann sum, Riemann sum, Riemann integral conditions for Riemann integrability, measure zero set, oscillation of a function, necessary and sufficient conditions for Riemann integrability (the Riemann-Lebesgue Theorem).
Section 6-2, Some Properties and Applications of the Riemann Integral, includes the topics: linearity of Riemann integral,
additivity of Riemann integral, properties of Riemann integral, Mean Value Theorem for Integrals, antiderivatives are continuous,
Fundamental Theorem of Calculus, change of variables, integration by parts, improper integrals, natural logarithm and exponential functions.
Chapter 8. Sequences and Series of Functions: Section 8-1, Sequences of Functions, includes the topics:
pointwise limit of a sequence of functions, a sequence of Riemann integrable functions can have a non-Riemann integrable pointwise limit, uniform limit, uniform limit of sequence of continuous functions is continuous, Uniform Convergence Theorem (a uniformly convergent sequence of Riemann integrable functions is itself Riemann integrable).
SUPPLEMENTAL REFERENCES: ome interesting videos on general mathematics are the following.
SYLLABUS ATTACHMENT: You can find an on-line version of the university's syllabus attachment (which contains general information concerning advisement, honor codes, dropping, etc.; accessed 1/16/2025).
ZOOM AND REMOTE ATTENDANCE: A ZOOM meeting is set up for each lecture through D2L. I encourage you to attend the in-person lectures if possible, but if you need to attend though ZOOM then that is fine. In particular, if you are unwell please consider the Zoom-alternative to attending class in person. You can ask questions through D2L and I will respond, just like in class. These ZOOM meetings will be recorded and posted on D2L.
CAMPUS SAFETY: East Tennessee State University is dedicated to creating a safe and healthy environment for all students, faculty, staff, and visitors by fostering a strong culture of safety that extends beyond compliance with regulations. All members of the ETSU community play a crucial role in this shared responsibility. You are encouraged to report any issue without fear, ensuring a supportive environment. As Buccaneers, make safety a priority and contribute to a positive, productive campus community by:
IMPORTANT DATES: (see the official ETSU calendar for more details; accessed 1/16/2025):
ROUGH SCHEDULE: We'll try to adhere to the following schedule... this may have to be revised.
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1-2. Properties ℝ as Ordered Field |
1.2.1(a), 1.2.3 |
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1-3. The Axiom of Completeness. |
1.3.4(b), 1.3.5 |
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Return to Dr. Bob's webpage
Last updated: August 31, 2025.