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SUBJECT : MATHEMATICS

REAL ANALYSIS

Unit II

Properties of real numbers - order, completeness, least upper bound property, Cauchy sequences

Welcome to our deep dive into the fundamental properties of real numbers. These properties are the bedrock upon which all of calculus and higher mathematics is built. Understanding them thoroughly is crucial for excelling in Real Analysis. We will explore the concepts of order, completeness, the least upper bound property, and finally, Cauchy sequences.

The Field Axioms and Order Axioms of Real Numbers

The set of real numbers, denoted by R, forms a complete ordered field. This means it satisfies two sets of axioms: the field axioms and the order axioms, along with a crucial completeness axiom.

Field Axioms:

These axioms define the basic arithmetic operations of addition (+) and multiplication (·) on the set of real numbers.

  • Closure: For any two real numbers a and b, a + b is a real number, and a · b is a real number.
  • Associativity: For any a, b, c in R, (a + b) + c = a + (b + c) and (a · b) · c = a · (b · c).
  • Commutativity: For any a, b in R, a + b = b + a and a · b = b · a.
  • Identity Elements: There exist unique elements 0 and 1 in R such that for any a in R, a + 0 = a and a · 1 = a. (Note: 0 ≠ 1)
  • Inverse Elements: For every a in R, there exists an element -a in R such that a + (-a) = 0. For every non-zero a in R, there exists an element a-1 (or 1/a) in R such that a · a-1 = 1.
  • Distributivity: For any a, b, c in R, a · (b + c) = (a · b) + (a · c).

Order Axioms:

These axioms define the concept of "less than" (<) on the set of real numbers, establishing its ordered nature.

  • Trichotomy: For any a, b in R, exactly one of the following is true: a < b, a = b, or a > b.
  • Transitivity: If a < b and b < c, then a < c.
  • Monotonicity of Addition: If a < b, then for any c in R, a + c < b + c.
  • Monotonicity of Multiplication: If a < b and c > 0, then a · c < b · c. If a < b and c < 0, then a · c > b · c.

From these axioms, we can derive many familiar properties, such as: if a < b and c < d, then a + c < b + d. Also, the square of any non-zero real number is positive. For instance, if x ≠ 0, then x2 > 0. This is because if x > 0, then x · x > 0 · x = 0. If x < 0, then x · x > 0 · x = 0 (since multiplying by a negative number reverses the inequality).

Completeness Axiom (Least Upper Bound Property)

This is the most distinctive axiom that separates the real numbers from the rational numbers. It ensures that there are "no gaps" in the number line.

Definitions:

  • Upper Bound: A real number u is an upper bound for a non-empty set S of real numbers if s ≤ u for all s ∈ S.
  • Lower Bound: A real number l is a lower bound for a non-empty set S of real numbers if s ≥ l for all s ∈ S.
  • Bounded Above: A set S is bounded above if it has an upper bound.
  • Bounded Below: A set S is bounded below if it has a lower bound.
  • Bounded: A set S is bounded if it is both bounded above and bounded below.

Least Upper Bound (Supremum) and Greatest Lower Bound (Infimum):

  • Least Upper Bound (sup S): If a non-empty set S is bounded above, then the least upper bound (or supremum) of S is the smallest real number M such that s ≤ M for all s ∈ S. This means that if M' is any other upper bound for S, then M ≤ M'.
  • Greatest Lower Bound (inf S): If a non-empty set S is bounded below, then the greatest lower bound (or infimum) of S is the largest real number m such that s ≥ m for all s ∈ S. This means that if m' is any other lower bound for S, then m' ≤ m.

The Completeness Axiom (Least Upper Bound Property): Every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in R.

Similarly, every non-empty set of real numbers that is bounded below has a greatest lower bound (infimum) in R. This is often called the Monotone Convergence Property when applied to sequences.

Key Takeaway: The real numbers are "complete" in the sense that there are no "holes" or "gaps." This property distinguishes them from the rational numbers. For example, the set of rational numbers S = {q ∈ Q | q2 < 2} is bounded above (e.g., by 2), but it does not have a least upper bound within the set of rational numbers. The supremum is √2, which is irrational. The completeness axiom guarantees that such a supremum exists within the real numbers.

Examples:

  1. Let S = {1, 2, 3}. The upper bounds are any real number ≥ 3. The least upper bound is 3. So, sup(S) = 3.
  2. Let S = {x ∈ R | x < 5}. The set is bounded above by any number ≥ 5. The least upper bound is 5. So, sup(S) = 5. Note that 5 is not in the set S.
  3. Let S = {1 - 1/n | n ∈ N}. This set is {0, 1/2, 2/3, 3/4, ...}. This set is bounded above by 1. The least upper bound is 1. sup(S) = 1.
  4. Let T = {-1, -2, -3}. The lower bounds are any real number ≤ -3. The greatest lower bound is -3. So, inf(T) = -3.
  5. Let T = {x ∈ R | x > 0}. This set is bounded below by 0. The greatest lower bound is 0. So, inf(T) = 0. Note that 0 is not in the set T.

Cauchy Sequences

Cauchy sequences provide an alternative way to characterize completeness, especially useful in more abstract metric spaces. A sequence (xn) is called a Cauchy sequence if, for any arbitrarily small positive number ε, there exists a natural number N such that for all integers m, n > N, the distance between xm and xn is less than ε.

Definition of a Cauchy Sequence:

A sequence of real numbers (xn) = (x1, x2, x3, ...) is a Cauchy sequence if for every ε > 0, there exists a natural number N such that |xm - xn| < ε for all m, n ≥ N.

In simpler terms, the terms of a Cauchy sequence get arbitrarily close to each other as the sequence progresses.

Relationship between Convergent and Cauchy Sequences:

For sequences of real numbers, convergence and being a Cauchy sequence are equivalent properties. This is a direct consequence of the completeness of R.

  • Theorem: Every convergent sequence of real numbers is a Cauchy sequence.
  • Theorem: Every Cauchy sequence of real numbers is convergent.

Therefore, for real numbers, the completeness axiom can be stated equivalently as: "Every Cauchy sequence of real numbers converges to a real number."

Proof Sketch (Convergent implies Cauchy): If (xn) converges to L, then for any ε/2 > 0, there is an N such that for n ≥ N, |xn - L| < ε/2. For m, n ≥ N, we have |xm - xn| = |xm - L + L - xn| ≤ |xm - L| + |L - xn| < ε/2 + ε/2 = ε. Thus, (xn) is Cauchy.
Proof Sketch (Cauchy implies Convergent - relies on Completeness): This direction is more involved and relies heavily on the completeness axiom. The idea is to construct a convergent subsequence and then show that the entire sequence must converge to the same limit. A common method involves constructing intervals based on the sequence terms and using the Boundedness Theorem (derived from completeness) and the Nested Interval Theorem.

Examples of Cauchy Sequences:

  1. The sequence (1/n) = (1, 1/2, 1/3, ...) is Cauchy. For any ε > 0, choose N = 1/ε. Then for m, n ≥ N, we have |1/m - 1/n| ≤ 1/m + 1/n ≤ 1/N + 1/N = 2/N = 2ε. Wait, this is not quite right. Let's refine. If m ≥ n ≥ N, then |1/m - 1/n| = 1/n - 1/m ≤ 1/n ≤ 1/N. So, choose N = 1/ε. Then for m, n ≥ N, assume m ≥ n without loss of generality. Then |1/m - 1/n| = 1/n - 1/m ≤ 1/n ≤ 1/N = ε. So, (1/n) is Cauchy. It converges to 0.
  2. Consider the sequence xn = Σk=0n (1/k!). This sequence is increasing. Let's check if it is bounded. We know e = Σk=0∞ (1/k!). The partial sums are bounded above by e. Since it's an increasing sequence bounded above, by the completeness axiom, it converges. Since it converges, it must be a Cauchy sequence.

Example of a Non-Cauchy Sequence:

Consider the sequence xn = (-1)n, i.e., (-1, 1, -1, 1, ...). Let ε = 1. Can we find an N such that |xm - xn| < 1 for all m, n ≥ N? If we choose m odd and n even (both ≥ N), then xm = -1 and xn = 1. So, |xm - xn| = |-1 - 1| = |-2| = 2. Since 2 is not less than 1, the condition |xm - xn| < ε is not met for ε = 1. Therefore, (xn) = ((-1)n) is not a Cauchy sequence. This sequence also does not converge.

Properties Derived from Completeness

The completeness axiom leads to several powerful theorems in real analysis.

Nested Interval Theorem:

If we have a sequence of closed intervals [an, bn] such that [an+1, bn+1] ⊆ [an, bn] for all n, and bn - an → 0 as n → ∞, then the intersection ∩n=1∞ [an, bn] contains exactly one point.

This theorem is a direct consequence of completeness. Consider the set S = {an | n ∈ N}. This set is non-empty and bounded above by any bk. By completeness, it has a supremum, say x. One can then show that x is the unique point in the intersection.

Bolzano-Weierstrass Theorem:

Every bounded infinite subset of R has at least one limit point (or accumulation point). A point p is a limit point of a set S if every neighborhood of p contains a point of S different from p.

This theorem is also proven using the completeness axiom. It implies that every bounded sequence of real numbers has a convergent subsequence. This is a fundamental tool for proving convergence.

Mnemonic for Limit Point: Think of a limit point as a point that the set "clusters around." If you zoom in infinitely close to a limit point, you'll always find more points from the set.

Intermediate Value Theorem (IVT):

If f is a continuous function on a closed interval [a, b], and y is any value between f(a) and f(b), then there exists at least one number c in (a, b) such that f(c) = y.

The proof of the IVT typically uses the completeness axiom, often by constructing a set of points where the function is less than or equal to y and finding its supremum.

Conclusion on Properties of Real Numbers:

The order axioms, combined with the completeness axiom (Least Upper Bound Property), define the unique structure of the real number system. This structure guarantees that concepts like limits, continuity, and convergence, which are central to calculus and analysis, behave predictably and consistently. Cauchy sequences provide an alternative, powerful perspective on completeness, linking the idea of terms getting close to each other with the guarantee of convergence. Mastering these foundational properties is key to understanding the rigorous development of mathematical analysis.

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