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Index of Chapters
First Semester
Introduction
Real Numbers
Beginning Limits
Differentiable and Continuous Functions on
R
Metric Spaces (Further Limits)
Compact Sets
Second Semester
Special Functions
Integration
Uniform Convergence
Countable and Uncountable Sets
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-Weierstrauss 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 Weierstrauss M-test
Convergence of power series
A nowhere differentiable function
The Weierstrauss 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