Schramm M.J. Introduction to Real Analysis

Dover Publications, Inc., 2008. — 384 p. — ISBN13: 978-0-486-46913-3.This text is an introductory course in real analysis, intended for students who have completed a calculus sequence. It is also designed to serve as preparation for advanced mathematics courses of many sorts. Though there are occasional references to it in exercises, linear algebra is not specifically a prerequisite for this text. Nevertheless, the changing role of linear algebra in the undergraduate curriculum is one of the main reasons this book comes to be the way it is. In the past, a first course in linear algebra was generally considered to be the place where one "learned to do proofs." The mathematics curriculum has gradually changed, though, and proofs as such are no longer the main focus of the typical linear algebra course. As a result, a student's first extensive experience with the logical and organizational skills necessary for the successful construction of proofs is often delayed until they find themselves in courses in which success is predicated on possession of those very skills.Building Proofs
Finite, Infinite, and Even Bigger
Algebra of the Real Numbers
Ordering, Intervals, and Neighborhoods
Upper Bounds and Suprema
Nested Intervals
Cluster Points
Topology of the Real Numbers
Sequences
Sequences and the Big Theorem
Compact Sets
Connected Sets
Series
Uniform Continuity
Sequences and Series of Functions
Differentiation
Integration
Interchanging Limit Processes
Increasing Functions
Continuous Functions and Differentiability
Continuous Functions and Integrability
We Build the Real Numbers