Isaac Newton

Augustin-Louis Cauchy

Karl Weierstrass

Georg Friedrich Bernhard Riemann

Images from Keith Lynn's "Pictures of Mathematicians" webpage and the The MacTutor History of Mathematics archive.

COURSE: MATH 4217/5217 Call # 33141/33158

TIME AND PLACE: 11:30-12:25 MWF in Gilbreath 304

INSTRUCTOR: Dr. Robert Gardner OFFICE HOURS: 10:25-11:20 MWF

OFFICE: Room 308G of Gilbreath Hall

PHONE: 439-6977 (308G Gilbreath), Math Department Office 439-4349

E-MAIL: gardnerr@etsu.edu
WEBPAGE: www.etsu.edu/math/gardner/gardner.htm (see my webpage for a copy of this course syllabus and updates for the course).

TEXT: An Introduction to Analysis, 2nd edition, by J. R. Kirkwood, Published by PWS Publishing Company and Waveland Press, Inc. 1995.

PREREQUISITES: It is assumed that each student has some experience with proof proving (at the level of MATH 2800 - Math Reasoning, for example). Of course, you should feel comfortable with references to results from freshman calculus.

ABOUT THE COURSE: In this course, we give a rigorous development of calculus and a study of the topology of the real line. Several of the results which we will 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. I will rely on the following sources:

1. The Elements of Real Analysis, by Robert Gardner Bartle. This text includes the results of our text along with sections on n-dimensional space, Rⁿ.
2. Topology, a First Course, by J. R. Munkres. A readable introduction to general topology. This text has been used in the past in our graduate Topology class (MATH 5350).
3. Real Analysis, by H. L. Royden. This is a standard text for a first graduate course in real analysis and is the text used at ETSU in graduate Analysis (MATH 5210). It includes the more advanced topics of measure theory, Lebesgue integration and L^p spaces.

GRADING: Homework (H) will be assigned and collected regularly. We will have two tests (T₁ and T₂). Your average will be computed as follows:

The tests will cover:

• Chapter 1 = Sets and Functions, Real Numbers as a Field, Completeness Axiom.
• Chapter 2 = Sequences of Real Numbers, Subsequences, Bolzano-Weierstrass Theorem.
• Chapter 3 = Topology of Real Numbers.
• Chapter 4 = Limits and Continuity, Monotone and Inverse Functions.
• Chapter 5 = Differentiation, Mean Value Theorems.

FINAL: There will be a comprehensive final on Monday December 9 from 3:50 to 5:50.

• Problem 1. Compare the 9 topologies given in class on X={a,b,c}.
• Problem 2. The limit of a sequence need not be unique in a topological space.
• Problem 3. (a) In a topological space, a finite union of closed sets is closed.
  (b) In a topological space, an arbitrary collection of closed sets is closed.

• G-1. The symmetric difference between sets A and B is defined as:
  
  (a) Prove...
  (b) Prove...
  (c) Prove...
• G-2. Extend 1.1.13c and 1.1.13d to a collection of indexed sets (see the note on page 13).
• G-3. Prove that for real a and b, we have |a+b|=|a|+|b| if an only if ab is nonnegative.
• G-4. Prove that if |X|=n, then the cardinality of the power set, P(X), is 2ⁿ.
• G-5. 2.2.10.
• G-6. 3.1.15a,b,c.
• G-7. 4.1.10.

• Bonus 1. A number is algebraic if it is the zero of P(x) where P(x) is a polynomial and the coefficients of P(x) are integers. Prove that the set of algebraic numbers is countable. HINT: Show that the collection of polynomials with integer coefficients is countable.
• Bonus 2. 2.3.11
• Bonus 3. Give an example of a topology on R other than the discrete, usual or trivial topoology.
• Bonus 4. Do another one from 4.1.1-4.1.12 (other than those in Homework number 17).
• Bonus 5. Do another one from 4.1.12-4.1.31 (other than those in Homework number 18).
• Bonus 6. Find a nowhere continuous function with the intermediate value property and explain.
• Bonus 7. Do another one from 4.2.1-4.2.15 (other than those in Homework number 19).

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