List of results

Theorems and lemmas 23 Definitions 8 Examples 13 Exercises 50

Ordered Fields and the Least Upper Bound Axiom

1.1 Ordered Fields and the Gap in the Rationals

Definition 1.1.1 Ordered Field
Lemma 1.1.1 Field Algebra
Example The rationals and the reals
Example A minimal change: the integers
Theorem 1.1.2 Rules for Inequalities
Definition 1.1.2 Absolute Value
Proposition 1.1.3 Rules for Absolute Value
Example Reading an absolute-value inequality as an interval
Theorem 1.1.4 Triangle Inequality
Exercise A1
Exercise B1: Three more rules
Exercise B2: Inserting a third point
Exercise B3: Solving inequalities
Exercise C1: Ordered fields contain the counting numbers
Exercise C2: Max and min without cases

1.2 Upper Bounds, Maxima and Suprema

Definition 1.2.1 Upper and Lower Bounds
Definition 1.2.2 Bounded Sets
Definition 1.2.3 Maximum and Minimum
Definition 1.2.4 Supremum
Theorem 1.2.1 Uniqueness of the Supremum
Definition 1.2.5 Infimum
Example A finite set, and a half-open interval
Example The supremum need not belong to the set
Example A supremum we cannot yet prove
Theorem 1.2.2 No Rational Supremum for the Rationals Below the Square Root of Two
Exercise A1
Exercise B1: Maxima are suprema
Exercise B2: Negating the definitions
Exercise B3: A supremum that is approached but not attained
Exercise C1: The two sides of the gap
Exercise C2: The least element of the upper bounds

1.3 The Least Upper Bound Axiom

Axiom 1.3.1 Least Upper Bound Axiom
Example A set of reals that must have a supremum
Theorem 1.3.2 Infimum by Reflection
Theorem 1.3.3 Greatest Lower Bound Property
Exercise A1
Exercise B1: The mirror identity
Exercise B2: The infimum is below the supremum
Exercise C1: The axiom in other ordered fields
Exercise C2: The upper bounds form a ray

1.4 Working with the Supremum

Theorem 1.4.1 Epsilon Characterization of the Supremum
Example Confirming a supremum with epsilon
Corollary 1.4.2 Sup Approximation
Theorem 1.4.3 Monotonicity of the Supremum
Theorem 1.4.4 Scaling the Supremum
Theorem 1.4.5 Supremum of a Sum
Exercise A1
Exercise B1: A computed supremum
Exercise B2: Supremum of a union
Exercise B3: Supremum of a difference
Exercise C1: Products of sets
Exercise C2: Exit check: sums of sets
Exercise C3: Find the wrong line

1.5 The Archimedean Property

Definition 1.5.1 Inductive set
Example Every nonzero natural number is at least 1
Theorem 1.5.1 Archimedean Property
Corollary 1.5.2
Example The supremum is 1
Example A non-Archimedean field
Exercise A1
Exercise B1: An infimum
Exercise B2: Scaled Archimedes
Exercise B3: Smaller than every 1/n
Exercise B4: Putting a proof in order
Exercise C1: Exit check: the Archimedean property, and a circular proof
Exercise C2: The order on the rational functions
Exercise C3: Archimedean without the axiom

1.6 Density of the Rationals and the Existence of Roots

Theorem 1.6.1 The Square Root of Two Exists
Theorem 1.6.2 Existence of Roots
Example Locating a cube root
Lemma 1.6.3 Integer Part
Theorem 1.6.4 Density of the Rationals
Example Running the proof
Corollary 1.6.5
Exercise A1
Exercise B1: Every real number is a supremum of rationals
Exercise B2: The square root of three, finished
Exercise B3: The cube root of five
Exercise C1: Exit check: the square root of two, in both fields
Exercise C2: Dyadic rationals
Exercise C3: Find the faulty line

1.7 Nested Intervals and the Uncountability of the Reals

Theorem 1.7.1 Nested Interval Theorem
Theorem 1.7.2 Uncountability of the Reals
Exercise A1
Exercise B1: Shrinking by a fixed factor
Exercise B2: An intersection that is not a point
Exercise C1: Exit check: nested intervals and the axiom
Exercise C2: Uncountable pieces
Exercise C3: Nested intervals in the rational functions
Exercise C4: A root of a cubic, from half a proof
Exercise R1
Exercise R2
Exercise R3
Exercise R4