Math 327 - Introduction to Real Analysis
See UW General Catalog for course description and prerequisite information.
Textbooks
- Advanced Calculus (Second Edition) by Patrick M. Fitzpatrick
- Principles of Mathematical Analysis (Third Edition) by Walter Rudin
- Elementary Analysis (Second Edition) by Kenneth A. Ross
Suggested Syllabus
The Real Number System
- sets, functions, equivalence relations
- fields, ordered fields, and their properties
- infimum and supremum
- Archimedean property
- intervals, absolute value and the triangle inequality
- density of the rationals
Sequences
- definition, convergence and limits
- Sandwich (Squeeze) theorem
- monotone sequences
- subsequences and sequential compactness
- liminf and limsup
- Caucy sequences and the completeness of the real numbers
Continuous Functions
- definition of continuity
- Extreme Value Theorem
- Intermediate Value Theorem
- monotone functions and inverses
- liminf and limsup
- uniform continuity
Series
- convergence and Cauchy criterion, absolute convergence
- Cauchy Condensation Theorem
- comparison, root and ratio tests
- alternating series
- liminf and limsup
- re-arrangements of terms of series