Real Number

Three axioms

1.The Field Axioms

These axioms define the algebraic structure of real numbers, making them a Field. For any :

• Closure: and are also real numbers.

• Commutative Law: and .

• Associative Law: and .

• Distributive Law: .

• Identity Elements:

  • There exists an additive identity such that .

  • There exists a multiplicative identity such that .

• Inverse Elements:

  • Every has an additive inverse such that .

  • Every has a multiplicative inverse such that .

2. The Order Axioms

These axioms establish the “greater than” () relationship, making an Ordered Field. For any :

• Trichotomy: Exactly one of the following is true: , , or .

• Transitivity: If and , then .

• Monotonicity of Addition: If , then .

• Monotonicity of Multiplication: If and , then .

3. The Completeness Axiom

Also known as the Continuity Axiom, this is the defining characteristic that separates real numbers from rational numbers. It is most commonly stated as:

The Least Upper Bound Axiom (LUB Axiom)

Definition: Every non-empty set of real numbers that is bounded above has a least upper bound (also called a Supremum) in .

Six Fundamental Theorems of Completeness

1. The Supremum Principle (Least Upper Bound Axiom)

• Statement: Every non-empty set of real numbers that is bounded above has a least upper bound (Supremum).

2. Monotone Convergence Theorem

• Statement: Every monotone (either non-decreasing or non-increasing) and bounded sequence of real numbers is convergent.

3. Nested Intervals Theorem (Nested Intervals Property)

• Statement: For a sequence of closed intervals such that each interval contains the next () and their lengths tend to zero, there exists exactly one point contained in all intervals.

4. Bolzano-Weierstrass Theorem

• Statement: Every bounded sequence has a convergent subsequence.

5. Cauchy Convergence Criterion

• Statement: A sequence of real numbers converges if and only if it is a Cauchy sequence.

• Definition of Cauchy Sequence: A sequence where the terms become arbitrarily close to each other as the index increases ().

6. Heine-Borel Theorem (Borel-Lebesgue Finite Covering Theorem)

• Statement: Every open cover of a closed and bounded interval has a finite subcover.

The Logical Loop of Proofs

1. Supremum Principle Monotone Convergence Theorem

• Logic: If a sequence is monotonic (e.g., increasing) and bounded above, the set of its terms has a Supremum () by the Supremum Principle.

• Result: Using the definition of Supremum, we can prove that the sequence must converge to .

2. Monotone Convergence Theorem Nested Intervals Theorem

• Logic: Consider a sequence of nested closed intervals .

• Result: The sequence of left endpoints is increasing and bounded, while is decreasing and bounded. By the Monotone Convergence Theorem, both sequences must converge. Since the interval length shrinks to zero, they must converge to the same point.

3. Nested Intervals Theorem Bolzano-Weierstrass Theorem

• Logic: This uses the Bisection Method.

• Result: For any bounded sequence, we can infinitely divide the interval containing it. By picking the sub-intervals that contain infinitely many terms, we create a Nested Interval stack that “traps” a specific point, which becomes the limit of a convergent subsequence.

4. Bolzano-Weierstrass Theorem Cauchy Convergence Criterion

• Logic: First, prove that every Cauchy sequence is bounded.

• Result: By B-W, this bounded sequence has a convergent subsequence. Because the sequence is “Cauchy” (terms get closer to each other), the entire sequence is forced to follow that subsequence to its limit.

5. Cauchy Convergence Criterion Heine-Borel Theorem

• Logic: Usually proven by Contradiction.

• Result: If a closed interval cannot be covered by a finite subcover, we can bisect it to find a smaller interval that also lacks a finite subcover. Repeating this creates a Cauchy sequence of points converging to a point . However, must be covered by some open set in the original cover, creating a contradiction in its neighborhood.

6. Heine-Borel Theorem Supremum Principle

• Logic: If we assume a bounded set has no Supremum, we can construct an open cover of the entire region where the Supremum “should” be.

• Result: By Heine-Borel, a finite number of these sets would cover the region, but this would lead to a logical contradiction regarding the boundary of the set.

Dedekind Cut

1. Definition

A Dedekind Cut is a method of partitioning the set of rational numbers into two disjoint, non-empty sets (the lower class) and (the upper class) such that:

1. Exhaustion: .

2. Order: Every element in is less than every element in (if and , then ).

3. No Greatest Element: contains no largest rational number. (This ensures each real number is represented by exactly one cut).

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2. The Three Types of Cuts

When you “cut” the rational number line, three logical scenarios can occur:

• Type 1: Rational Cut

  The “gap” falls exactly on a rational number.

  • Example: , . This cut represents the number 2.

• Type 2: Irrational Cut (The Breakthrough)

  The “gap” falls where no rational number exists.

  • Example: , .

  • Observation: has no maximum and has no minimum within . Dedekind defined this “hole” as the irrational number .

• Type 3: Impossible Cut

  A gap where has a maximum and has a minimum. This is impossible in due to its density (you can always find a rational between any two others).

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3. Supremum Principle starting from the Dedekind Cut axiom

Theorem: Every non-empty set that is bounded above has a least upper bound (Supremum).

Proof:

1. Let be the set of all rational numbers such that is an upper bound of .

2. Let be the set of all other rational numbers ().

3. The pair forms a Dedekind Cut because it partitions , is ordered, and has no maximum.

4. By the Dedekind Axiom, this cut defines a real number .

5. By definition of the cut, is the smallest element of the upper bounds, thus .