Week 1 Notes: Real Numbers, Sequences, and Fundamental Limits

Real Numbers: Field Axioms and Order Axioms

Notation
- N := {1, 2, 3, …}, the set of natural numbers
- Z := {0, ±1, ±2, ±3, …}, the set of integers
- Q := {p/q : p, q ∈ Z, q ≠ 0}, the set of rational numbers
- R := the set of real numbers
- Geometric view: R is described by a bijection with points on a line extending infinitely in both directions
Field Axioms: binary operations + and · on R satisfy the following for all a, b, c ∈ R
1) a + b ∈ R
2) a + (b + c) = (a + b) + c
3) ∃0 ∈ R such that a + 0 = a = 0 + a
4) ∃−a ∈ R such that a + (−a) = 0 = (−a) + a
5) a + b = b + a (commutativity of addition)
6) a · b ∈ R
7) a · (b · c) = (a · b) · c
8) ∃1 ∈ R such that a · 1 = a = 1 · a (multiplicative identity)
9) ∀a ≠ 0, ∃ a⁻¹ ∈ R with a · a⁻¹ = 1 = a⁻¹ · a (multiplicative inverses)
10) a · b = b · a (commutativity of multiplication)
11) distributive laws: a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c
Order Axioms (structure of positivity in R)
- There exists a nonempty subset R⁺ ⊆ R of positive real numbers such that for all a, b ∈ R⁺:
  1) a + b ∈ R⁺
  2) a · b ∈ R⁺
  3) For any a ∈ R, exactly one of the following holds: a ∈ R⁺, a = 0, −a ∈ R⁺
Definition 1 (ordering via subtraction)
- Let a, b ∈ R. If a − b ∈ R⁺, then a > b.
- If a − b ∈ R⁺ ∪ {0}, then a ≥ b.
Law of Trichotomy
- For any a, b ∈ R, exactly one of: a > b, a = b, a < b.

Absolute value and related properties

Definition 2 (Absolute value)
- For x ∈ R, |x| =
  $\begin{cases} x, & \text{if } x > 0, \ 0, & \text{if } x = 0, \ -x, & \text{if } x < 0. \end{cases}$
- Examples: |7| = 7, |−3| = 3; |−x| = |x|
Properties of |·|
- (1) |ab| = |a| · |b|
- (2) |a + b| ≤ |a| + |b|
- (3) ||a| − |b|| ≤ |a − b|
Distance interpretation
- |a − b| denotes the distance between real numbers a and b
- |a| denotes the distance of a from the origin
ε-translation notes
- |x − a| < ε translates to a − ε < x < a + ε
- |x − a| > ε translates to x > a + ε or x < a − ε

Bounds, upper/lower bounds, and supremum/infimum

Definition 3 (Bounds)
- Bounded above: ∃M ∈ R such that x ≤ M for all x ∈ S; M is an upper bound of S
- Bounded below: ∃m ∈ R such that x ≥ m for all x ∈ S; m is a lower bound of S
- Bounded set: S is bounded if it is both bounded above and bounded below; otherwise S is unbounded
- If S ⊆ R is bounded, ∃K ∈ R such that |x| ≤ K for all x ∈ S
Examples of sets (A–E)
- A = {5, 8, 10, 11, 12, 15, 19, 37, 99}
- B = {x ∈ R : x < 2}
- C = {x ∈ R : x ≥ 5}
- D = {x ∈ R : 1 < x < 3}
- E = {x ∈ R : 1 ≤ x ≤ √2}
Observations
- A, D, E are bounded
- B is bounded above but not bounded below
- C is bounded below but not bounded above
- N is bounded below but not bounded above
- Z, Q, R are neither bounded above nor bounded below
Maximum and minimum
- M is the maximum of S if M is an upper bound of S and M ∈ S
- m is the minimum of S if m is a lower bound of S and m ∈ S

Supremum, Infimum, and Completeness of R

Definition 4 (Supremum and Infimum)
- Least Upper Bound (lub) or Supremum: If S is bounded above, ℓ is a supremum of S if ℓ is an upper bound of S and ℓ ≤ v for every upper bound v of S
- Greatest Lower Bound (glb) or Infimum: If S is bounded below, g is an infimum of S if g is a lower bound of S and g ≥ v for every lower bound v of S
- Supremum and Infimum are unique if they exist; denote sup(S), inf(S)
Examples of sup/inf for given sets
- sup(A) = 99, inf(A) = 5
- sup(B) = 2, inf(B) = −∞ (B is unbounded below)
- sup(D) = 3, inf(D) = 1
- sup(E) = √2, inf(E) = 1
Remark on maximum/minimum vs sup/inf
- A set may have a supremum without a maximum (e.g., non-attaining upper bound)
- D and B illustrate this phenomenon in various ways
Completeness Property of R
- Every nonempty bounded above subset of R has a supremum in R
Completeness vs Q
- Q is a field and satisfies the order axioms, but Q is not complete
- Example S = {x ∈ Q : x² ≤ 2} is bounded above in Q, but sup(S) = √2 ∉ Q
- Therefore, there exist nonempty bounded-above subsets of Q without a supremum in Q
Exercise: Prove that √2 ∉ Q (√2 is irrational)
Proof (sketch)
- Suppose √2 = p/q with integers p, q, gcd(p, q) = 1, q ≠ 0
- Then p² = 2 q², so p² is even → p is even → p = 2 p1
- Substitution gives q² = 2 p1² → q is even
- Contradicts gcd(p, q) = 1
- Hence √2 is irrational

Greatest Integer Function and Intervals

Definition 5 (Greatest integer function)
- For any x ∈ R, there exists a unique m ∈ Z with x − 1 < m ≤ x
- Denote by [x] the greatest integer ≤ x
- Examples: [3.859] = 3, [−3.581] = −4, [2.9] = 2, [2.01] = 2, [4] = 4
Definition 6 (Intervals)
- Closed interval (bounded): [p, q] = {x ∈ R : p ≤ x ≤ q}
- Unbounded closed interval: [p, ∞) = {x ∈ R : p ≤ x}, (−∞, q] = {x ∈ R : x ≤ q}
- Open interval (bounded): (p, q) = {x ∈ R : p < x < q}
- Unbounded open interval: (p, ∞) = {x ∈ R : p < x}, (−∞, q) = {x ∈ R : x > q}
- Semi-open/semi-closed: [p, q) = {x ∈ R : p ≤ x < q}, (p, q] = {x ∈ R : p < x ≤ q}
Consequences of Completeness (selected)
- Infimum of a nonempty set bounded below exists (via the negative set trick)
- Archimedean Property: For any x ∈ R, ∃ n ∈ N with n > x
- Between any two real numbers there exists a rational and an irrational number
- Proof sketch: Given a < b, b − a > 0. 1/(b − a) ∈ R. By Archimedean property ∃ n ∈ N with n > 1/(b − a). Then ∃ m ∈ Z with na < m < nb, hence a < m/n < b (rational in (a, b)). For irrational, consider a + √2 and b + √2 and find rational c with a + √2 < c < b + √2, hence a < c − √2 < b (c rational, c − √2 irrational).

Real Sequences

Definition 7 (Real Sequence)
- A sequence is a function f: N → R, f(n) = xn; denote as (xn)
Representations of sequences
- (1) Listing: e.g., (1, 4, 8, 11, 52, …)
- (2) Rule: x_n = 3 n^2
- (3) Recurrence: x1 = 4, x{n+1} = 2 x_n − 5
Some example sequences (illustrative)
- (a) a_n = 2 (constant sequence)
- (b) a_n = 1 − (−1)^n / (2n)
- (c) a_n = 1 + (−1)^n (oscillating)
- (d) an defined by a1 = 1, a{n+1} = an + a_{n−1} (Fibonacci-type)
Definition 8 (Convergent sequence)
- (xn) converges to ℓ ∈ R if ∀ε > 0 ∃ m ∈ N such that |xn − ℓ| < ε for all n ≥ m
- Denote lim{n→∞} xn = ℓ
Example: x_n = 1/n
- For any ε > 0, ∃ m ∈ N with |1/n − 0| < ε for all n ≥ m
- Archimedean property: ∃ m with m > 1/ε, so |1/m| < ε
- Therefore 1/n → 0
Theorem 2.1 (Uniqueness of limit)
- If xn → ℓ1 and xn → ℓ2, then ℓ1 = ℓ2
- Sketch: For any ε > 0, choose m1, m2 with |xn − ℓ1| < ε/2 and |xn − ℓ2| < ε/2 for n ≥ max(m1, m2). Then |ℓ1 − ℓ2| ≤ |xn − ℓ1| + |xn − ℓ2| < ε, hence ℓ1 = ℓ2
Examples continued
- If a ∈ R and xn ≡ a, then xn → a
- If an = 2/(n^2 + 1), then |an| ≤ 2/n^2 → 0, so a_n → 0
Remark: Divergence and convergence notions
- If a sequence does not converge to any real number, it diverges
- Classic divergent example: a_n = (−1)^n
Definition 9 (Bounded sequence)
- A sequence (xn) is bounded if {x1, x2, …} is bounded, i.e., ∃M > 0 with |xn| ≤ M for all n
- If not bounded, the sequence is unbounded
Examples of bounded/unbounded sequences
- (xn) = (−1)^n and (yn) = 1/n are bounded
- (an) = n and (bn) = n^2/(n+1) are unbounded

Basic convergence properties and the Sandwich Principle

Theorem 2.3 (Convergent ⇒ bounded)
- If (xn) → ℓ, then ∃ε > 0 such that ∃m with |xn − ℓ| < ε for n ≥ m, hence x_n is bounded
Remark 2.4 (Converse false)
- There exist bounded sequences that do not converge (e.g., (−1)^n)
Remark 2.5
- If a sequence is unbounded, it cannot converge
Theorem 2.6 (Algebraic limits of convergent sequences)
- If xn → a and yn → b, then:
  1) xn + yn → a + b
  2) xn − yn → a − b
  3) xn yn → ab
  4) xn / yn → a/b if b ≠ 0
  5) If xn ≤ yn for all n, then a ≤ b
Theorem 2.7 (Sandwich Theorem)
- If there exist sequences an ≤ cn ≤ bn for all large n and lim an = lim bn = c, then lim cn = c
Remark 2.8 (Dominated sub-sequences and limits)
- If |bn| ≤ cn for all n ≥ m and cn → 0, then bn → 0
Example class: convergence of geometric-like sequences
- Let a ∈ R and an = a^n. Then (an) converges iff −1 < a ≤ 1
- Cases:
- a = 0: a_n ≡ 0 → 0
- a = 1: a_n ≡ 1 → 1
- a = −1: a_n oscillates between −1 and 1 (not convergent)
- 0 < |a| < 1: set r = 1/|a| > 1; then |a|^n = 1/r^n → 0 (via binomial/estimates)
- |a| > 1: sequence is unbounded (diverges)
Additional example type (limit to 0)
- a_n = (n^3 + 3n^2 + 4n^4 + 8n^2 + 2)/n^k (illustrative forms in notes)
- Another common example: an = (1/n) sin(1/n) with 0 ≤ |an| ≤ 1/n → 0

Monotone sequences and convergence criteria

Definition 10 (Monotone sequences)
- A sequence is increasing if an ≤ a{n+1} for all n
- A sequence is decreasing if an ≥ a{n+1} for all n
- A sequence is monotonic if it is either increasing or decreasing
- Examples: an = 1/n is decreasing; bn = (n+1)/n is increasing; (−1)^n is not monotonic
Theorem 2.9 (Monotone Convergence Criterion)
- An increasing sequence that is bounded above converges to sup{a_n : n ∈ N}
- Sketch of proof: use well-ordering (supremum) and the definition of boundedness to show x_n → s
Remark 2.10 (Decreasing sequences)
- A decreasing sequence that is bounded below converges to inf{a_n : n ∈ N}
Example (Babylonian iteration for √2)
- Let a1 = 3/2 and a{n+1} = 1/2 (an + 2/an)
- All a_n > 0, so the sequence is bounded below by 0
- a{n+1} − an = (2 − an^2)/(2 an)
- One can show a_n is decreasing for all n and bounded below by 0, hence convergent
- If an → ℓ, then ℓ satisfies ℓ = (ℓ + 2/ℓ)/2 ⇒ 2ℓ^2 = ℓ^2 + 2 ⇒ ℓ^2 = 2 ⇒ ℓ = √2 (positive sequence), so an → √2

Summary connections and implications

Real numbers are a complete ordered field: every nonempty set bounded above has a supremum, and every nonempty set bounded below has an infimum
Completeness distinguishes R from Q; Q is not complete
The Archimedean property ensures there are integers between any two reals after appropriate scaling, underlines convergence proofs like 1/n → 0
Convergence, boundedness, and monotonicity are foundational for analysis, providing tools like the Sandwich Theorem and Monotone Convergence Criteria for proving limits
The examples illustrate how to apply definitions to concrete sequences and how to reason about limits, convergence, and divergence