Analysis
Preface
1
Starting at the begining: the natural numbers
1.1
The Peano axioms
1.2
Addition
1.3
Multiplication
1.4
Vocabulary
1.5
Foot Note
2
Set theory
2.1
Fundamentals
2.2
Russell’s paradox
2.3
Functions
2.4
Images and inverse images
2.5
Cartesian products
2.6
Cardinality of sets
2.7
Vocabulary
3
Integers and rationals
3.1
The integers
3.2
The rationals
3.3
Absolute value and exponentiation
3.4
Gaps in the rational numbers
4
The real numbers
4.1
Cauchy sequences
4.2
Equivalent Cauchy sequences
4.3
The construction of the real numbers
4.4
Ordering the reals
4.5
The least upper bound property
4.6
Real exponentiation, part I
4.7
Foot Notes
5
Limits of sequences
5.1
Convergence and limte laws
5.2
The Extended real number system
5.3
Suprema and Infima of sequences
5.4
Limsup, Liminf, and limit points
5.5
Subsequences
5.6
Real exponentiation, part II
6
Series
6.1
Finite series
6.2
Infinite series
6.3
Sums of non-negative numbers
6.4
Rearrangement of series
6.5
The root and ratio tests
7
Infinite sets
7.1
Countability
7.2
Summation on infinite sets
7.3
Uncountable sets
7.4
The axiom of choice
7.5
Ordered sets
8
Continuous functions on
\(\mathbb{R}\)
8.1
Subsets of the real line
8.2
The algebra of real-valued functions
8.3
Limiting values of functions
8.4
Continuous functions
8.5
Left and right limits
8.6
The maximum principle
8.7
The intermediate value theorem
8.8
Monotonic functions
8.9
Uniform continuity
8.10
Limits at infinity
9
Differentiation of functions
9.1
Basic definitions
9.2
Local maxima, local minima, and derivatives
9.3
Monotone functions and derivatives
9.4
Inverse functions and derivatives
9.5
L’Hôpital’s rule
10
The Riemann integral
10.1
Partitions
10.2
Piecewise constant functions
10.3
Upper and lower Riemann integrals
10.4
Basic properties of the Riemann integrals
10.5
Riemann integrability of continuous functions
10.6
Riemann integrability of monotone functions
10.7
A non-Riemann integrable function
10.8
The Riemann-Stieltjes integral
10.9
The two fundamental theorems of calculus
10.10
Consequences of the fundamental theorems
11
Appendix: the basics of mathematical logic
11.1
Mathematical statements
11.2
Implication
11.3
The stucture of proofs
11.4
Variables and quantifiers
11.5
Nested quantifiers
11.6
Equality
11.7
Vocabulary
12
Appendix: the decimal system
12.1
The decimal representation of natural numbers
12.2
The decimal representation of real numbers
12.3
Vocabulary
References
Published with bookdown
Analysis
Analysis
Neoneu
2024-06-26
Preface
This is a note of
Analysis
by Terence Tao.