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Real Analysis:
Math 4050-4060
Valerio De Angelis
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Front Matter
How is this course different from Calculus?
1
Introduction
1.1
The various types of numbers
1.1.1
\(\mathbb N, \mathbb Z\)
and
\(\mathbb Q\)
1.1.2
Order
1.1.3
Incompleteness of
\(\mathbb Q\)
1.2
Logic and set theory
1.2.1
Statements
1.2.2
Logic
1.2.3
Quantifiers
1.2.4
Sets
1.2.5
Set operations
1.2.6
Cardinality
1.2.7
De Morgan’s laws
1.2.8
Cartesian product
1.3
Functions
1.3.1
Basic definitions
1.3.2
Properties of functions
1.4
Basic facts about the integers
1.4.1
Even and Odd integers
1.4.2
Divisibility
1.4.3
The Well-Ordering Principle
1.5
Writing proofs
1.5.1
Direct proofs
1.5.2
Contrapositive
1.5.3
Contradiction
1.5.4
Induction
1.5.5
Cases in proofs
1.6
Exercises
2
The Real Numbers
2.1
Ordered Fields
2.1.1
The field axioms
2.1.2
Uniqueness questions
2.1.3
Axiomatic proofs
2.1.4
Subtraction and division
2.1.5
The order axioms
2.1.6
Absolute value and inequalities
2.1.7
Intervals
2.2
The Completeness Axiom
2.2.1
Maximum and minimum
2.2.2
Upper and lower bounds
2.2.3
Bounded sets
2.2.4
Supremum and infimum
2.2.5
The completeness axiom
2.2.6
The Archimedean principle
2.2.7
Equivalent forms
2.2.8
Density of
\(\mathbb Q\)
2.3
Exercises
3
Sequences
3.1
Definition and notation
3.1.1
Examples
3.1.2
Properties of sequences
3.2
Limits
3.2.1
Definition of limit
3.2.2
Examples
3.2.3
Convergent sequences are bounded
3.2.4
Proving that a sequence diverges
3.2.5
Properties of limits
3.3
Some basic theorems
3.3.1
Sequences in closed intervals
3.3.2
The squeeze theorem
3.3.3
A bounded, monotone sequence converges
3.4
Infinite limits
3.4.1
Examples
3.5
limsup and liminf
3.5.1
Tail of a sequence
3.5.2
limsup and liminf
3.6
Subsequences
3.6.1
Monotone subsequences
3.6.2
The Bolzano Weierstrass theorem
3.7
Exercises
4
Functions
4.1
Limits
4.1.1
Definition
4.1.2
Properties of limits
4.1.3
Basic results on limits
4.1.4
One sided limits
4.1.5
Limits at infinity
4.1.6
Infinite limits
4.2
Monotone functions
4.3
Continuity
4.3.1
Types of discontinuity
4.3.2
The Intermediate Value Theorem
4.3.3
The Extreme Value Theorem
4.4
Invertible functions
4.5
Uniform continuity
4.6
Exercises
5
Differentiation
5.1
The derivative
5.1.1
Definition of derivative
5.1.2
Basic results
5.2
Local extrema
5.3
Rolle’s theorem
5.4
The Mean Value Theorem
5.4.1
The Monotonicity Theorem
5.5
The Power Rule
5.6
Exercises
6
Integration
6.1
Riemann sums
Backmatter
Colophon
Colophon
This book was authored in PreTeXt.
