Math 341-Real Analysis I

Course Description:  This course is a first introduction to Real Analysis.  "Real" refers to the real numbers.
Much of our work will revolve around the real number system.  We will start by carefully considering the
axioms that describe it.  Students will be asked to consider many functions that take on real values---that is,
each object in our domain will be associated with a real number.  For instance, every point in the plane can
be associated with its distance from the origin.  Two points in the plane give rise to a real number: the
distance betweent them.  The concept of distance will be a major theme of the course.
"Analysis" is one of the principle branches of mathematics.  One often hears that analysis is the theoretical
underpinnings of the calculus, but though this has a kernel of truth, it is an answer that misleads by
oversimplifying.  Certainly, analysis had its inception in the attempt to give a careful, mathematically sound,
explanation of the ideas of the calculus.  But over the last century, analysis has grown out of its original
packaging and is now much more than simply the theory of the calculus. Analysis is the mathematics of
"closeness"--- the mathematics of limiting processes.  The idea of continuity can be phrased in terms of limits.
Both derivatives and integrals are the end results of taking a limit.  Compactness is a property of sets that
underlies many of the most important theorems encountered in calculus.  These and related ideas will be the
subject of the course.  Prerequisites: Math 213 and Math 222 or permission of the instructor.