Unit I: Real Numbers and Their Topological Properties

1. Algebraic Properties of ℝ

The set of real numbers, denoted by ℝ, forms an algebraic structure known as a field under the operations of addition (+) and multiplication (·). This means that ℝ satisfies the following field axioms:

These two operations are connected by the Distributive Law:

For all a, b, c ∈ ℝ, a · (b + c) = (a · b) + (a · c).

Derived Algebraic Properties

From these basic axioms, we can derive several fundamental algebraic properties that are frequently used in proofs:

• Uniqueness of Identity and Inverse: The additive identity (0) and multiplicative identity (1) are unique. For any element a, its additive inverse (-a) and multiplicative inverse (a^-1, for a ≠ 0) are also unique.
• Multiplication by Zero: For any a ∈ ℝ, a · 0 = 0.
• Cancellation Laws: If a + c = b + c, then a = b. If a · c = b · c and c ≠ 0, then a = b.
• Sign Rules: (-1) · a = -a, and (-a) · (-b) = a · b.

2. Order Properties of ℝ

The real numbers possess an inherent order structure, making ℝ an ordered field. There is a non-empty subset P of ℝ, called the set of positive real numbers, that satisfies the following Positivity Axioms:

1. If a, b ∈ P, then a + b ∈ P (Closure under addition).
2. If a, b ∈ P, then a · b ∈ P (Closure under multiplication).
3. If a ∈ ℝ, then exactly one of the following holds: a ∈ P, a = 0, or -a ∈ P (Law of Trichotomy).

We use these axioms to define inequality relations on ℝ:

• We write a > b (or b < a) if a - b ∈ P.
• We write a ≥ b (or b ≤ a) if a - b ∈ P or a - b = 0.

Key Rules of Inequalities

Important Observation: If a ≠ 0, then a² > 0. This is because if a ∈ P, then a · a ∈ P (a² > 0). If -a ∈ P, then (-a) · (-a) ∈ P (a² > 0). As a consequence, 1 > 0 (since 1 = 1²).

3. Absolute Value and the Real Line

The absolute value of a real number represents its distance from zero on the real number line, regardless of direction.

Definition: For any real number x, the absolute value of x, denoted by |x|, is defined as:
|x| = x if x ≥ 0
|x| = -x if x < 0

Properties of Absolute Value

• Non-negativity: |x| ≥ 0 for all x ∈ ℝ, and |x| = 0 if and only if x = 0.
• Multiplicativity: |x · y| = |x| · |y| for all x, y ∈ ℝ.
• Bounds: -|x| ≤ x ≤ |x| for all x ∈ ℝ.
• Triangle Inequality: For all x, y ∈ ℝ, |x + y| ≤ |x| + |y|.
• Reverse Triangle Inequality: For all x, y ∈ ℝ, ||x| - |y|| ≤ |x - y|.

Step-by-Step Proof of Triangle Inequality:

We know that -|x| ≤ x ≤ |x| and -|y| ≤ y ≤ |y|. Adding these inequalities together yields:

-(|x| + |y|) ≤ x + y ≤ |x| + |y|

By properties of inequalities, this is equivalent to |x + y| ≤ |x| + |y|.

Geometric Interpretation of Distance

For any two points x, y ∈ ℝ, the value d(x, y) = |x - y| is the distance between x and y on the real line. Thus, the inequality |x - a| < d means that the distance from x to a is strictly less than d.

4. ε-Neighbourhood of a Point

The concept of a neighbourhood is fundamental to real analysis, bridging algebraic inequalities and topology.

Definition: Let a ∈ ℝ and ε > 0. The ε-neighbourhood of a (denoted by V_ε(a)) is the set of all real numbers whose distance from a is less than ε:
V_ε(a) = {x ∈ ℝ : |x - a| < ε}.

Geometrically, V_ε(a) is the open interval (a - ε, a + ε) centered at a with radius ε.

Key Theorem on Neighbourhoods

An important characterization of a point belonging to a set is stated as follows:

A point x ∈ ℝ belongs to V_ε(a) if and only if:
a - ε < x < a + ε

Observation: If a real number x satisfies x ≥ a - ε and x ≤ a + ε for every ε > 0, then x = a. This is an essential technique for proving two real numbers are equal.

5. Boundedness of Sets

Sets of real numbers can be characterized by whether they are constrained by upper or lower limits.

Definitions

• Bounded Above: A set S ⊂ ℝ is bounded above if there exists a real number u ∈ ℝ such that x ≤ u for all x ∈ S. The number u is called an upper bound of S.
• Bounded Below: A set S ⊂ ℝ is bounded below if there exists a real number w ∈ ℝ such that x ≥ w for all x ∈ S. The number w is called a lower bound of S.
• Bounded: A set S ⊂ ℝ is bounded if it is both bounded above and bounded below. Equivalently, S is bounded if there exists a positive real number M > 0 such that |x| ≤ M for all x ∈ S.
• Unbounded: A set S is unbounded if it is not bounded. This means for any M > 0, there exists some x ∈ S such that |x| > M.

Examples of Boundedness

6. Least Upper Bound (Supremum) and Greatest Lower Bound (Infimum)

A set can have infinitely many upper or lower bounds. We are particularly interested in the most restrictive of these bounds.

Least Upper Bound (Supremum)

Definition: Let S be a non-empty subset of ℝ that is bounded above. A number u ∈ ℝ is called the supremum (or least upper bound, lub) of S, written as u = sup S, if it satisfies:

1. u is an upper bound of S (i.e., x ≤ u for all x ∈ S).
2. If v is any other upper bound of S, then u ≤ v.

Greatest Lower Bound (Infimum)

Definition: Let S be a non-empty subset of ℝ that is bounded below. A number w ∈ ℝ is called the infimum (or greatest lower bound, glb) of S, written as w = inf S, if it satisfies:

1. w is a lower bound of S (i.e., x ≥ w for all x ∈ S).
2. If t is any other lower bound of S, then t ≤ w.

The ε-Characterization of Supremum and Infimum

In analytical proofs, we use equivalent ε-definitions that do not rely on comparing all bounds:

• Supremum ε-Criterion: u = sup S if and only if:
  1. x ≤ u for all x ∈ S.
  2. For every ε > 0, there exists at least one element x_ε ∈ S such that x_ε > u - ε.
• Infimum ε-Criterion: w = inf S if and only if:
  1. x ≥ w for all x ∈ S.
  2. For every ε > 0, there exists at least one element x_ε ∈ S such that x_ε < w + ε.

Common Mistake: Confusing the supremum/infimum with the maximum/minimum of a set. The maximum (or minimum) of a set must be an element of the set. The supremum and infimum do not need to belong to the set. For example, for S = (0, 1), sup S = 1 and inf S = 0, but S has no maximum and no minimum element.

7. Completeness Property of ℝ

The key distinguishing feature between the rational numbers (ℚ) and the real numbers (ℝ) is the property of completeness. The rational numbers have "gaps", whereas the real numbers form a continuous, uninterrupted line.

The Completeness Property of ℝ (Supremum Property):
Every non-empty subset of ℝ that is bounded above has a supremum in ℝ.

By symmetry, this also implies that every non-empty subset of ℝ that is bounded below has an infimum in ℝ.

Why the Rational Numbers (ℚ) are Not Complete

Consider the set of rational numbers:

S = {r ∈ ℚ : r > 0 and r² < 2}

This set S is non-empty (since 1 ∈ S) and bounded above in ℚ (for example, by 2 ∈ ℚ). However, S does not have a supremum in ℚ. The supremum of S on the real line is the irrational number √2, which is not in ℚ. Thus, ℚ fails the completeness property.

8. Boundedness of Functions

The concept of boundedness extends from sets to functions based on their range.

Definition: Let f: D → ℝ be a function defined on a set D.

• f is bounded above on D if there exists M ∈ ℝ such that f(x) ≤ M for all x ∈ D.
• f is bounded below on D if there exists m ∈ ℝ such that f(x) ≥ m for all x ∈ D.
• f is bounded on D if there exists M > 0 such that |f(x)| ≤ M for all x ∈ D. This means the range f(D) is a bounded set in ℝ.

Supremum and Infimum of Functions

If a function f: D → ℝ is bounded, we can define its supremum and infimum over its domain D:

• sup_{x ∈ D} f(x) = sup {f(x) : x ∈ D}
• inf_{x ∈ D} f(x) = inf {f(x) : x ∈ D}

Example: Let f: (0, ∞) → ℝ be defined by f(x) = 1/x.

• f is bounded below by 0, since 1/x > 0 for all x > 0. Thus, inf f(x) = 0.
• f is not bounded above because as x approaches 0, f(x) becomes arbitrarily large. Hence, f is an unbounded function.

9. The Archimedean Property

The Archimedean Property asserts that the set of natural numbers ℕ is not bounded above in ℝ. This implies that there are no "infinitely large" real numbers and no "infinitely small" non-zero real numbers.

Archimedean Property Theorem: If x ∈ ℝ, then there exists an n_x ∈ ℕ such that n_x > x.

Proof using Completeness:

Suppose the statement is false. Then there exists some x ∈ ℝ such that n ≤ x for all n ∈ ℕ. This means the set of natural numbers ℕ is bounded above by x. By the Completeness Property, ℕ must have a supremum in ℝ, say u = sup ℕ.

Using the ε-characterization of supremum, for ε = 1, there must exist m ∈ ℕ such that:

m > u - 1

Adding 1 to both sides gives m + 1 > u. Since m ∈ ℕ, it follows that m + 1 ∈ ℕ. This contradicts the assumption that u is an upper bound of ℕ. Thus, our initial assumption must be false, proving that ℕ is unbounded above.

Crucial Corollaries

These equivalent formulations of the Archimedean Property are frequently used in limits and analysis:

1. If y > 0 and x > 0, there exists n ∈ ℕ such that ny > x.
2. For any ε > 0, there exists n ∈ ℕ such that 1/n < ε.
3. If x > 0, there exists n ∈ ℕ such that n - 1 ≤ x < n.

10. Density Theorem

The density theorem shows how tightly rational and irrational numbers are woven throughout the real line.

Density of Rational Numbers: If x and y are real numbers with x < y, then there exists a rational number r ∈ ℚ such that x < r < y.

Proof Outline:

Since x < y, we have y - x > 0. By the Archimedean Property, there exists n ∈ ℕ such that 1/n < y - x, which means ny - nx > 1.

Applying the Archimedean Property again, we can find an integer m ∈ ℤ such that m - 1 ≤ nx < m. This implies m ≤ nx + 1 < ny.

Combining these inequalities gives nx < m < ny. Dividing by n, we get x < m/n < y. Setting r = m/n, we have found a rational number r such that x < r < y.

Density of Irrational Numbers: If x and y are real numbers with x < y, then there exists an irrational number z ∈ ℝ ‚ ℚ such that x < z < y.

Explanation: Since x < y, we also have x/√2 < y/√2. By the density of rationals, there exists a non-zero rational number r ∈ ℚ such that x/√2 < r < y/√2. Multiplying by √2 yields x < r√2 < y. Since r is rational and √2 is irrational, z = r√2 is irrational. Thus, an irrational number exists between any two real numbers.

11. Intervals and Their Characterization

Intervals are geometrically connected subsets of the real line. They are defined using the order properties of ℝ.

Types of Intervals

Characterization Theorem of Intervals

How do we mathematically define an interval without relying on explicit endpoints? We use the "intermediate value" property of sets:

Characterization Theorem: A non-empty subset S of ℝ is an interval if and only if for every x, y ∈ S with x < y, any point z satisfying x < z < y also belongs to S.

This means if a set contains no gaps, it must be an interval.

12. Nested Interval Property

The Nested Interval Property is another formulation of the completeness of ℝ, stated in terms of a sequence of shrinking closed intervals.

Nested Interval Property: Let I_n = [a_n, b_n], n ∈ ℕ, be a sequence of closed bounded intervals such that they are nested, meaning:
I₁ ⊇ I₂ ⊇ I₃ ⊇ ... ⊇ I_n ⊇ I_n+1 ⊇ ...
Then, there exists at least one real number ξ ∈ ℝ that belongs to all these intervals, i.e., ξ ∈ ∩_n=1^∞ I_n.

Step-by-Step Proof/Intuition:

Because the intervals are nested, the sequence of left endpoints is increasing and bounded above by any right endpoint: a₁ ≤ a₂ ≤ a₃ ≤ ... ≤ b_n ≤ ... ≤ b₁.

Let A = {a_n : n ∈ ℕ} be the set of left endpoints. Since A is non-empty and bounded above by any b_n, by the Completeness Property of ℝ, it has a supremum. Let ξ = sup A.

Since ξ is the supremum of left endpoints, we have a_n ≤ ξ for all n. Also, because any b_n is an upper bound of A, we must have ξ ≤ b_n for all n. Thus, for every n ∈ ℕ:

a_n ≤ ξ ≤ b_n

This means ξ ∈ [a_n, b_n] = I_n for all n, proving that ξ ∈ ∩_n=1^∞ I_n.

Uniqueness Condition

If the lengths of the intervals, b_n - a_n, satisfy inf {b_n - a_n : n ∈ ℕ} = 0, then the common point ξ is unique.

Warning: The Nested Interval Property fails if the intervals are not closed. For example, consider the nested open intervals I_n = (0, 1/n) for n ∈ ℕ. Their intersection is empty (∩_n=1^∞ I_n = ∅) because there is no positive real number x such that 0 < x < 1/n for all n (by the Archimedean Property).