MAT 3751: Real Analysis II
Winter 2006
Instructor:
Dr. Brian Gill
Office:
OMH 209
E-mail:
bgill@spu.edu
Phone:
281-2954
Web page:
http://myhome.spu.edu/bgill
University and Departmental Mission:
Seattle Pacific University seeks to be a premier Christian university fully
committed to engaging the culture and changing the world by graduating people of competence and character, becoming
people of wisdom, and modeling grace-filled community. The mathematics department at Seattle Pacific University seeks to
provide excellent instruction to enable our students to be competent in the mathematics required for their chosen fields, and
to share our expertise with the community through service and leadership. Hence, common goals for students in mathematics
courses include 1) becoming competent in the topics covered in the course, 2) demonstrating skills and attitudes which
contribute to professional, ethical behavior, 3) the ability to communicate mathematically, in both written and verbal form,
and 4) learning to appreciate the beauty and utility of mathematics.
Course Overview:
What is analysis? Analysis is the branch of mathematics that deals with properties of functions. Real
analysis deals primarily with real-valued functions of real numbers; essentially it is the theory behind elementary calculus.
This course provides an introduction to the topics of analysis: functions, cardinality, topology of the real line, and limits.
The emphasis of the rest of the course will be on real numbers and the foundations of real analysis. We will define and
construct the set of real numbers and study various properties of sets of real numbers. In addition, we will learn the precise,
formal definition of a limit of a sequence of real numbers and study properties of these limits. These topics lay the
foundation for a rigorous study of functions of a real variable, including limits, continuity, derivatives, and integrals.
Learning Objectives:
By the end of the course, you should:
Know and be able to work with basic properties of the real number system, including:
o the completeness axiom and its consequences; and
o basic topology of the real line, including open and closed sets and compactness.
Know, understand, and be able to apply the following definitions and theorems:
o the
N
ε
−
definition of the limit of a sequence;
o the algebraic and order limit theorems;
o the monotone convergence theorem;
o the Bolzano-Weierstrass theorem;
o the Cauchy criterion;
o the
ε
δ
−
definitions of continuity and the limit of a function;
o the intermediate value theorem;
o the definitions of pointwise and uniform convergence of sequences of functions;
o the formal definition of a derivative; and
o the mean value theorem.
Be able to write formal proofs relating to the topics listed above.
Have an increased familiarity with and understanding of many topics first encountered in calculus, including
basic properties of sequences, series, limits, continuity, and derivatives.
Understand some of the historical reasons for replacing intuitive notions of limits with the precise modern
definitions.
In addition to the specific content oriented objectives above, this course will help you to:
Sharpen your analytical and critical reasoning skills and improve your ability to express mathematical ideas with
clarity and coherence.
Attain a greater appreciation of the need for precise definitions, careful reasoning, and close argument in
mathematics.
Prepare for further upper division mathematics courses and greater abstraction.
Finally, I hope that you will have fun accomplishing these objectives, even if the material is difficult and takes a lot of
time and effort.
Prerequisites:
Successful completion of MAT 3749 (Introduction to Analysis) prior to starting this course.
Textbook:
Understanding Analysis by Stephen Abbott, Springer-NY, 2001. ISBN: 0387950605
Software:
The course will include several lab activities which require the use of the mathematical software package Maple.
The software is available for your use in all of the open computer labs on campus. If you wish to use the software on your
own computer, a downloadable student version of Maple can be purchased for $75 (details will be provided in class).
