Comps Syllabus for Analysis

Download analysis comps syllabus

Mathematical Induction

The Real Numbers

  • Rational and irrational numbers
  • Real numbers and the axiom of completeness

Sequences

  • Convergence
  • Convergence of bounded monotone sequences
  • Cauchy sequences
  • The Bolzano-Weierstrass Theorem for sequences

Point-Set Theory

  • Limit points (also called cluster points or accumulation points)
  • The Bolzano-Weierstrass Theorem for sets
  • Open and closed sets
  • Compact sets and the Heine-Borel Theorem

Infinite series

  • Convergence
  • p-series and geometric series
  • Absolute and conditional convergence
  • Comparison, ratio, and alternating series tests

Limits and Continuity

  • The limit of a function
  • The definition of continuity and relation to sequences
  • Continuity of sums, products, quotients, and compositions
  • The Intermediate Value Theorem
  • Boundedness
  • Attainment of extreme values
  • Uniform continuity

Differentiability and Derivatives

  • Limit definition of derivative
  • Derivatives at local extreme points
  • The Mean Value Theorem and Rolle's Theorem

Sequences of Functions

  • Pointwise and uniform convergence
  • Continuity of the limit function
  • Proving uniform convergence

Series of Functions

  • Pointwise and uniform convergence
  • The Weierstrauss M-test
  • Power series
  • Radius and interval of convergence, behavior at endpoints
  • Continuity and differentiation of power series

Integration

  • Definition of the Riemann integral
  • Properties of the Riemann integral
  • Integrability of a continuous function over [a,b]
  • Integration of sequences and series
  • Integration of power series