Index of Chapters

First Semester

1. Introduction
2. Real Numbers
3. Beginning Limits
4. Differentiable and Continuous Functions on R
5. Metric Spaces (Further Limits)
6. Compact Sets

Second Semester

7. Special Functions
8. Integration
9. Uniform Convergence
10. Countable and Uncountable Sets
11. Construction of the Real Number System

Chapter 1: Introduction

• Mathematical rigor and standards of proof

• The need for axioms

Chapter 2: Real Numbers

• Inequalities, and algebra involving inequalities

• Properties of Ordered Sets

• Bounded sets

• The least upper bound (supremum) of a set

• The Least Upper Bound Property

• The axioms of the real numbers

• Basic facts about the real numbers:
  • The Archimedean Property
  • Density of the rationals in the reals
  • Irrational numbers
  • The greatest lower bound of a set
• The absolute value function

• Open sets of real numbers

• The extended real number system

Chapter 3: Beginning Limits

• Sequences of real numbers

• Convergence of sequences

• Algebra of convergent sequences

• n th roots of positive real numbers

• Sequences converging to infinity

• Convergence of monotone sequences

• Closed subsets of R; relationship between open and closed sets

Chapter 4: Differentiable and Continuous Functions on R

• Limits of functions

• Limits of functions in terms of limits of sequences

• Algebra involving limits of functions

• Continuous functions

• The exponential functions

• Differentiable functions

• Rules to simplify differentiation:
  • Sums and scalar multiples of differentiable functions
  • Product and quotient rules
  • Chain rule
• Limits of functions at infinity.

• Left and right limits of functions.

Chapter 5: Metric Spaces (Further Limits)

• Construction of the complex number system

• Examples of different types of convergence (Euclidean metric and the uniform metric)

• Definition of metric space

• Cauchy-Schwartz inequality

• Convergence and continuity in metric spaces

• Techniques for proving convergence and continuity in Euclidean spaces

• Open and closed sets in metric spaces

• Continuity and open sets

• Properties of open and closed sets

• Constructing and recognizing open and closed sets

• Sets which are both open and closed

• Connectedness

• Continuous functions and connected sets

• The Intermediate Value Theorem

• Logarithms

• Finding roots by the method of bisection

Chapter 6: Compact Sets

• Subsequences

• Bolzano-Weierstrass Theorem

• Sequential compactness

• Characterize sequentially compact sets in Euclidean spaces

• Properties of sequentially compact sets:
  • Continuous images of sequentially compact sets are sequentially compact
  • Continuous functions are bounded on sequentially compact sets
  • A continuous bijection from a sequentially compact set has a continuous inverse
  • Continuous functions attain their maximum on sequentially compact sets
  • Applications to differentiable functions:
    • Rolle's Theorem
    • The Mean Value Theorem
    • The inverse function theorem
  • Uniform continuity

• Compact sets(topological definition)

• Finite intersection property

• Compact and sequentially compact sets are the same in metric spaces

• Complete metric spaces

• Compact metric spaces are complete

Chapter 7: Special Functions

• Series

• Comparison test for series (an application of completeness)

• Power series and the disk/interval of convergence

• The "exponential" power series E(x) and the number e

• Power series are differentiable inside their disk of convergence

• E(x) is equal to e^x

• Derivatives of exponential, power and logarithmic functions

• Definition of sine and cosine

• Basic properties of trig functions (and the definition of Pi)

• The polar form for complex numbers and complex logarithms

• Taylor's theorem: approximating functions with partial sums of power series

• Estimating the numbers e and Pi

• Rearrangements of series and conditional convergence

• Double series

Chapter 8: Integration

• Definition of the Riemann Integral

• A non-integrable function

• Algebra of integration

• Fundamental Theorem of Calculus

• Inequalities involving integrals

• Riemann-Stieltjes Integration

Chapter 9: Uniform Convergence

• Definition of uniform convergence

• Uniform limits of continuous functions are continuous

• Uniform limits of Riemann integrable functions

• The Weierstrass M-test

• Convergence of power series

• A nowhere differentiable function

• The Weierstrass approximation theorem

Appendix A: Countable and Uncountable Sets

• When two sets have the same cardinality

• Countable sets and criteria for countability

• The countable union of countable sets is countable

• Uncountable sets

• The real line is uncountable

• Cantor's proof of the existence of transcendental numbers

Appendix B: Construction of the Real Numbers

• Existence

• Uniqueness

• Defining an abelian ring

• Defining a field

• Establishing well-defined order

• Establishing order

• Least upper bound property