The Real Numbers

What are the axioms of ℝ?

It is a complete ordered field meaning it has:

• Field axioms (you can add, subtract, multiply, divide by non-zero)

• Order axioms (you can compare any two reals)

• Completeness axioms (there are no gaps)

What are the field axioms of ℝ?

These are what makes ℝ a field

• Commutativity: a + b = b + a, a · b = b · a

• Associativity: (a + b) + c = a + (b + c), (a b)c = a(b c)

• Distributivity: a(b + c) = ab + ac

• Identities: 0 and 1 exist with a + 0 = a, a · 1 = a

• Additive inverse: for every a there exists –a with a + (–a) = 0

• Multiplicative inverse: for every a ≠ 0 there exists a⁻¹ with a · a⁻¹ = 1

What are the order axioms of ℝ?

Basically how < and > behave

• Trichotomy: exactly one of (a < b), (a = b), (a > b) is true.

• Transitivity: if a < b and b < c, then a < c.

• Addition rule: if a < b then a + c < b + c.

• Multiplication rule: if a < b and c > 0 then ac < bc.

What is the completeness axiom of ℝ?

This is the single property that makes ℝ different from ℚ. Completeness fills the gaps. Mathematically: Every non-empty set S ⊂ ℝ that is bounded above has a least upper bound (supremum) in ℝ. If there’s a ceiling on a set (no element of S goes above some number), there’s a smallest possible ceiling inside ℝ.

What are the consequences of completeness?

Because ℝ is complete:

• Every monotone bounded sequence has a limit in ℝ.

• Cauchy sequences converge in ℝ.

• The Intermediate Value Theorem and Extreme Value Theorem (from calculus) depend on this property.

Why is ℚ not complete?

Because some bounded-above subsets (e.g. {x ∈ ℚ | x² < 2}) have no least upper bound in ℚ; √2 ∉ ℚ.

What is a monotone bounded sequence?

“Monotone” means it never changes direction:

• Monotone increasing: each term ≥ the previous one (a₁ ≤ a₂ ≤ a₃ ≤ …)

• Monotone decreasing: each term ≤ the previous one (a₁ ≥ a₂ ≥ a₃ ≥ …)

“Bounded” means it stays within limits:

• Bounded above: there’s some ceiling M such that aₙ ≤ M for all n.

• Bounded below: there’s some floor m such that aₙ ≥ m for all n.

Every monotone and bounded sequence of real numbers converges to a limit in ℝ.

What is a cauchy sequence?

A cauchy sequence is one where, as you go further out in the list, the terms get closer and closer to each other — even if you don’t yet know what value they’re approaching.

Formally: (aₙ) is Cauchy if for every ε > 0, there exists N such that for all m, n ≥ N,
|a_n−a_m| < ε

What is the Intermediate Value Theorem?

If f is continuous on [a,b], and f(a) and f(b) are on opposite sides of some value L, then there exists c ∈ (a,b) such that f(c) = L. (aka a continuous function can’t “jump over” a value — it must hit every intermediate value)

(e.g. f(x) = x² – 2.
f(1) = –1, f(2) = 2.
Because f is continuous, there must be some c between 1 and 2 where f(c)=0 → that’s √2.)

What is the Extreme Value Theorem?

If f is continuous on a closed, bounded interval [a,b],
then f achieves a maximum and a minimum on that interval.

In symbols: ∃ xₘᵢₙ, xₘₐₓ ∈ [a,b] such that f(xₘᵢₙ) ≤ f(x) ≤ f(xₘₐₓ) for all x ∈ [a,b].

(e.g. f(x) = x² on [–2,3].
Minimum at x = 0 (f = 0), maximum at x = 3 (f = 9))

What is a supremum?

The smallest upper bound

What is an infimum?

The greatest lower bound

What does unbounded mean?

There’s no limit on one side