Analysis I

Contents

• 1.1 What is analysis?
• 1.2 Why do analysis?

2 Starting at the beginning:the natural numbers

• 2.1 The Peano axioms
• 2.2 Addition
• 2.3 Multiplication

3 Set theory

• 3.1 Fundamentals
• 3.2 Russell’s paradox (Optional)
• 3.3 Functions
• 3.4 Images and inverse images
• 3.5 Cartesian products
• 3.6 Cardinality of sets

4 Integers and rationals

• 4.1 The integers
• 4.2 The rationals
• 4.3 Absolute value and exponentiation
• 4.4 Gaps in the rational numbers

5 The real numbers

• 5.1 Cauchy sequences
• 5.2 Equivalent Cauchy sequences
• 5.3 The construction of the real numbers
• 5.4 Ordering the reals
• 5.5 The least upper bound property
• 5.6 Real exponentiation, part I

6 Limits of sequences

• 6.1 Convergence and limit laws
• 6.2 The Extended real number system
• 6.3 Suprema and Infima of sequences
• 6.4 Limsup, Liminf, and limit points
• 6.5 Some standard limits
• 6.6 Subsequences
• 6.7 Real exponentiation, part II

7 Series

• 7.1 Finite series
• 7.2 Infinite series
• 7.3 Sums of non-negative numbers
• 7.4 Rearrangement of series
• 7.5 The root and ratio tests

8 Infinite sets

• 8.1 Countability
• 8.2 Summation on infinite sets
• 8.3 Uncountable sets
• 8.4 The axiom of choice
• 8.5 Ordered sets

9 Continuous functions on R

• 9.1 Subsets of the real line
• 9.2 The algebra of real-valued functions
• 9.3 Limiting values of functions
• 9.4 Continuous functions
• 9.5 Left and right limits
• 9.6 The maximum principle
• 9.7 The intermediate value theorem
• 9.8 Monotonic functions
• 9.9 Uniform continuity
• 9.10 Limits at infinity

10 Differentiation of functions

• 10.1 Basic definitions
• 10.2 Local maxima, local minima, and derivatives
• 10.3 Monotone functions and derivatives
• 10.4 Inverse functions and derivatives
• 10.5 L’Hopital’s rule

11 The Riemann integral

• 11.1 Partitions
• 11.2 Piecewise constant functions
• 11.3 Upper and lower Riemann integrals
• 11.4 Basic properties of the Riemann integral
• 11.5 Riemann integrability of continuous functions
• 11.6 Riemann integrability of monotone functions
• 11.7 A non-Riemann integrable function
• 11.8 The Riemann-Stieltjes integral
• 11.9 The two fundamental theorems of calculus
• 11.10 Consequences of the fundamental theorems

A Appendix:the basics of mathematical logic

• A.1 Mathematical statements
• A.2 Implication
• A.3 The structure of proofs
• A.4 Variables and quantifiers
• A.5 Nested quantifiers
• A.6 Some examples of proofs and quantifiers
• A.7 Equality

B Appendix:the decimal system

• B.1 The decimal representation of natural numbers
• B.2 The decimal representation of real numbers
• Index
• Texts and Readings in Mathematics