The Order Axioms of ℝ Notes

1. Order axioms in R

The real numbers are more than a field—they are an ordered field.

Definition: A set is ordered if there exists a binary relation "<" satisfying:

Axioms of Order:

1. Trichotomy: For all

   $a, b \in ℝ$

   a,b∈R, exactly one of the following is true:

   • $a < b$

     a<b

   • $a = b$

     a=b

   • $a > b$

     a>b

2. Transitivity: If

   $a < b$

   a<b and

   $b < c$

   b<c, then

   $a < c$

   a<c

3. Addition Property: If

   $a < b$

   a<b, then

   $a + c < b + c$

   a+c<b+c

4. Multiplication Property: If

   $0 < a$

   0<a and

   $0 < b$

   0<b, then

   $0 < ab$

   0<ab

From these, we define:

• $a ≤ b$

  a≤b means

  $a < b$

  a<b or

  $a = b$

  a=b

• $a ≥ b$

  a≥b means

  $b ≤ a$

  b≤a

2. Intervals in R

Real analysis often works with intervals, which are sets of real numbers defined by inequalities.

Types of Intervals:

• Open interval:

  $(a, b) = \{ x \in ℝ \mid a < x < b \}$

  (a,b)={x∈R∣a<x<b}

• Closed interval:

  $[a, b] = \{ x \in ℝ \mid a \leq x \leq b \}$

  [a,b]={x∈R∣a≤x≤b}

• Half-open:

  $(a, b], [a, b)$

  (a,b],[a,b)

• Infinite:

  $(-\infty, b), (a, \infty), (-\infty, \infty)$

  (−∞,b),(a,∞),(−∞,∞)

These play an essential role in defining limits, continuity, and compactness.

2.1. Density of Q in R

Statement:

Between any two real numbers

a<b, there exists a rational number

q such that

a<q<b

Proof Sketch:

Let

a<b. Since

b−a>0, choose

n∈N such that

n1​<b−a. Then choose

m∈Z such that

nm​>a. Adjust

m to find a rational number

q=nm​∈(a,b)

Implication:

• The rationals are “everywhere” in the reals

• Rational approximations are always possible

• Fundamental in defining limits and continuity

2.2. Archimedean property in R

Statement:

For every

x∈R, there exists a natural number

n∈N such that

n>x.
Equivalently,

R has no infinitely large or infinitesimal elements.

Consequences:

• $\exists n \in ℕ$

  ∃n∈N such that

  $\frac{1}{n} < \epsilon$

  n1​<ϵ for any

  $\epsilon > 0$

  ϵ>0

• Ensures that ℕ is unbounded in ℝ

• Essential in constructing sequences that converge to zero

The Archimedean Property bridges the discrete world of natural numbers and the continuum of the real line.