Exploring the Infinite: An Introduction to Proof and Analysis

Fundamentals of Abstract Mathematics 
Basic Notions
A First Look at Some Familiar Number Systems 
Integers and natural numbers 
Rational numbers and real numbers
Inequalities 
A First Look at Sets and Functions
Sets, elements, and subsets 
Operations with sets
Special subsets of R: intervals
Functions 
Mathematical Induction 
First Examples 
Defining sequences through a formula for the n-th term 
Defining sequences recursively
First Programs 
First Proofs: The Principle of Mathematical Induction 
Strong Induction 
The Well-Ordering Principle and Induction 
Basic Logic and Proof Techniques 
Logical Statements and Truth Table
Statements and their negations 
Combining statements 
Implications 
Quantified Statements and Their Negations
Writing implications as quanti ed statements 
Proof Techniques 
Direct Proof 
Proof by contradiction 
Proof by contraposition 
The art of the counterexample
Sets, Relations, and Functions 
Sets 
Relations 
The definition 
Order Relations 
Equivalence Relations 
Functions 
Images and pre-images 
Injections, surjections, and bijections 
Compositions of functions 
Inverse Functions 
Elementary Discrete Mathematics
Basic Principles of Combinatorics 
The Addition and Multiplication Principles
Permutations and combinations 
Combinatorial identities 
Linear Recurrence Relations 
An example 
General results 
Analysis of Algorithms
Some simple algorithms 
Omicron, Omega and Theta notation 
Analysis of the binary search algorithm 
Number Systems and Algebraic Structures 
Representations of Natural Numbers
Developing an algorithm to convert a number from base
10 to base 2.
Proof of the existence and uniqueness of the base b representation of an element of N 
Integers and Divisibility 
Modular Arithmetic 
Definition of congruence and basic properties 
Congruence classes 
Operations on congruence classes 
The Rational Numbers 
Algebraic Structures 
Binary Operations 
Groups
Rings and fields 
Cardinality 
The Definition 
Finite Sets Revisited 
Countably Infinite Sets
Uncountable Sets 
Foundations of Analysis 
Sequences of Real Numbers
The Limit of a Sequence
Numerical and graphical exploration 
The precise de nition of a limit
 Properties of Limits
Cauchy Sequences 
Showing that a sequence is Cauchy 
Showing that a sequence is divergent 
Properties of Cauchy sequences 
A Closer Look at the Real Number System 
R as a Complete Ordered Field 
Completeness 
Why Q is not complete 
Algorithms for approximating square root 2
Construction of R 
An equivalence relation on Cauchy sequences of rational
numbers
Operations on R
Verifying the field axioms
Defining order 
Sequences of real numbers and completeness
Series, Part 1 
Basic Notions
Exploring the sequence of partial sums graphically and
numerically
Basic properties of convergent series 
Series that diverge slowly: The harmonic series
Infinite geometric series 
Tests for Convergence of Series 
Representations of real numbers 
Base 10 representation 
Base 10 representations of rational numbers
Representations in other bases 
The Structure of the Real Line 
Basic Notions from Topology
Open and closed sets 
Accumulation points of sets 
Compact sets 
Subsequences and limit points 
First definition of compactness 
The Heine-Borel Property 
A First Glimpse at the Notion of Measure 
Measuring intervals 
Measure zero 
The Cantor set 
Continuous Functions
Sequential Continuity
Exploring sequential continuity graphically and numerically
Proving that a function is continuous 
Proving that a function is discontinuous
First results
Related Notions 
The epsilon-delta□ condition 
Uniform continuity 
The limit of a function 
Important Theorems 
The Intermediate Value Theorem
Developing a root-finding algorithm from the proof of the
IVT
Continuous functions on compact intervals 
Differentiation
Definition and First Examples 
Properties of Differentiable Functions and Rules for Differentiation
Applications of the Derivative 
Series, Part 2 
Absolutely and Conditionally Convergent Series
The rst example 
Summation by Parts and the Alternating Series Test 
Basic facts about conditionally convergent series 
Rearrangements
Rearrangements and non-negative series 
Using Python to explore the alternating harmonic series
A general theorem 
A Very Short Course on Python 
Getting Stated 
Why Python? 
Python versions 2 and 3
Installation and Requirements 
Integrated Development Environments (IDEs) 
Python Basics 
Exploring in the Python Console 
Your First Programs
Good Programming Practice 
Lists and strings 
if . . . else structures and comparison operators 
Loop structures 
Functions 
Recursion