Math 327 - Introduction to Real Analysis

See UW General Catalog for course description and prerequisite information.


  • Advanced Calculus (Second Edition) by Patrick M. Fitzpatrick
  • Principles of Mathematical Analysis (Third Edition) by Walter Rudin

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


  • 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


  • 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

Instructors Only