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MATH508: Survey of Analysis


Revised: November 2006


Course Description

This course is a review of analysis at the advanced calculus level. This course places an emphasis on proof and is fairly wide in its scope. Our clients are mainly interested either in obtaining a master's degree in mathematics education or qualifying for a position in two-year college teaching. Three semester hours.


Objectives

  1. To develop a background in logic used in analysis.

  2. To study elementary properties of sets and functions.

  3. To understand the topology of the real numbers and the basic algebraic properties of the real numbers.

  4. To understand proofs and apply theorems in the following areas of advanced calculus: sequences, limits, continuity, differentiation and integration.


Text

Steven R. Lay. Analysis with an Introduction to Proof, Second Edition. Prentice-Hall, 1990.


Grading Procedure

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.


Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.


Course Outline
  • Chapter 1: Logic and Proof
    Logical connectives, Quantifiers, Techniques of Proof: I, Techniques of Proof: II,

  • Chapter 2: Sets and Functions
    Basic Set Operations, Relations, Functions, Cardinality, Axioms for Set Theory

  • Chapter 3: The Real Numbers
    Natural Numbers and Induction, Ordered Fields, The Completeness Axiom, Topology of the Reals, Compact Sets, Metric Spaces

  • Chapter 4: Sequences
    Convergence, Limit Theorems, Monotone Sequences and Cauchy Sequences, Subsequences

  • Chapter 5: Limits and Continuity
    Continuous Functions, Properties of Continuous Functions, Uniform Continuity, Continuity in Metric Spaces

  • Chapter 6: Differentiation
    The Derivative, The Mean Value Theorem, L'Hospital's Rule, Taylor's Theorem

  • Chapter 7: Integration
    The Riemann Integral, Properties of the Riemann Integral, The Fundamental Theorem of Calculus


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