Real Numbers Axioms

• Real numbers (R) are a set with addition (+) and multiplication (·) operations
• This set follows an ordering (

Axioms of Addition

• Associative Law: a + (b + c) = (a + b) + c for all real numbers a, b, c
• Commutative Law: a + b = b + a for all real numbers a, b
• Zero: There exists a real number 0 such that a + 0 = a for all real numbers a
• Additive inverse: For each real number a, there exists a real number -a such that a + (-a) = 0

Axioms of Multiplication

• Associative Law: a(bc) = (ab)c for all real numbers a, b, c
• Commutative Law: ab = ba for all real numbers a, b
• One: There exists a real number 1, where 1 ≠ 0, such that a · 1 = a for all real numbers a
• Multiplicative inverse: For each real number a ≠ 0, there exists a real number a⁻¹ such that a * a⁻¹ = 1

Distributive Law

• a(b + c) = ab + ac for all real numbers a, b, c

Order Axioms

• Trichotomy: For each real number a, exactly one of the following is true: a > 0, a = 0, or -a > 0
• If a > 0 and b > 0, then a + b > 0
• If a > 0 and b > 0, then ab > 0

Definition of Positivity

• Real number a is larger than b (a > b) if a - b > 0

Upper and Lower Bounds

• An upper or lower bound element does not have to be part of the set and is not unique
• If there is a number A ∈ R such that x ≤ A for all x ∈ S, then A is an upper bound of S, and S is bounded above
• If there is a number a ∈ R such that x ≥ a for all x ∈ S, then a is a lower bound of S, and S is bounded below
• If S is bounded above and below, then S is considered bounded

Max and Min

• Elements must be part of the set and is unique
• If S has an upper bound M that is an element of S, M is the greatest element (maximum) of S, written as M = max S
• If S has a lower bound m that is an element of S, m is the least element (minimum) of S, written as m = min S

Supremum

• Unique and can be outside of the set
• For a nonempty subset S of R, a real number M is the supremum (least upper bound) if:
  • M is an upper bound of S
  • If L is any upper bound of S, then M ≤ L; the supremum of S is denoted by sup S

Infimum

• Unique and can be outside of the set
• For a nonempty subset S of R, a real number m is the infimum (greatest lower bound) if:
  • m is a lower bound of S
  • If l is any lower bound of S, then m ≥ l; the infimum of S is denoted by inf S

Dedekind Completeness Axiom

• Every nonempty subset of R that is bounded above has a supremum

Peano's Axioms (Natural Numbers)

• 0 is a natural number (0 ∈ N)
• If a is a natural number, then a + 1 is a natural number
• For all natural numbers a, a + 1 is not equal to 0
• If S is a subset of N such that 0 ∈ S and a + 1 ∈ S for all a ∈ S, then S = N

Density

• A subset S of R is dense in R if for all x, y ∈ R with x < y, there exists an s ∈ S such that x < s < y

Real Sequences

• An ordered list of infinitely many real numbers a1, a2, a3, a4, .... is denoted by (an) = a1, a2, a3,. . . .
• The number an is the n-th term of the sequence and the subscript n is called the index

Convergence and Divergence

• For a sequence (an) and L ∈ R, the statement 'an tends to L as n tends to infinity', written as 'an → L as n → ∞', means: ∀ ε > 0 ∃ K ∈ R ∀n ∈ N, n ≥ K, |an − L| < ε
• If an → L as n → ∞, (an) converges to L, and is written as limn→∞ an = L
• A sequence (an) is convergent if it converges to some real number; otherwise, it is divergent

Bounding

• A sequence (an) is bounded above if there exists a number M ∈ R such that an ≤ M for all indices n
• A sequence (an) is bounded below if there exists a number m ∈ R such that an ≥ m for all indices n
• A sequence (an) is bounded if it is bounded above and bounded below

Sequence Characteristics

• Increasing: an ≤ an+1 for all indices n
• Strictly increasing: an < an+1 for all indices n
• Decreasing: an ≥ an+1 for all indices n
• Strictly decreasing: an > an+1 for all indices n
• Monotonic: either increasing or decreasing
• Strictly monotonic: either strictly increasing or strictly decreasing

Sequence to Infinity

• an tends to infinity as n tends to infinity (an → ∞ as n → ∞) if for every A ∈ R, there exists K ∈ R such that an > A for all n ≥ K
• an tends to minus infinity as n tends to infinity (an → −∞ as n → ∞) if for every A ∈ R, there exists K ∈ R such that an < A for all n ≥ K