The Greatest Lower Bound Property is a crucial concept in real analysis, defining the infimum of a set. It's closely tied to the completeness of real numbers and plays a key role in proving the existence of certain values and limits.
This property helps us understand how sets of numbers are bounded from below. It's essential for various mathematical proofs and applications, including optimization problems, integration, and the study of sequences and series in advanced calculus.
Greatest Lower Bound Property
Definition and Properties
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The greatest lower bound (GLB) of a set S of real numbers is a real number g that satisfies two conditions:
g is a lower bound of S, meaning g ≤ x for all x in S
g is greater than or equal to any other lower bound of S
The GLB is also called the infimum of the set, denoted as inf(S)
If a set S has a GLB, then the set S is bounded below
The GLB of a set S, if it exists, is unique
This means there cannot be two distinct GLBs for the same set
If a set S does not have a lower bound, then inf(S) = -∞
For example, the set of all negative real numbers {x ∈ ℝ : x < 0} does not have a lower bound, so its infimum is -∞
Examples of GLB
Consider the set S = {1, 2, 3, 4, 5}
The GLB of S is 1 because 1 ≤ x for all x in S, and 1 is the largest real number satisfying this condition
Let T = {x ∈ ℝ : x^2 > 4}
The GLB of T is 2 because 2 ≤ x for all x in T, and 2 is the largest real number satisfying this condition
For the set U = (0, 1], the GLB is 0
Note that 0 is not included in the set U, but it is still the GLB because it is the largest lower bound of U
The GLB of the empty set ∅ is defined as ∞
This is because every real number is a lower bound for the empty set, and there is no largest real number
Proof of Greatest Lower Bound Property
Proving the Existence of GLB
To prove the GLB property, consider a non-empty set S of real numbers that is bounded below
Define a set A = {x ∈ ℝ : x is a lower bound of S}
A is the set of all lower bounds of S
Show that A is non-empty and bounded above
A is non-empty because S is bounded below, so there exists at least one lower bound of S
A is bounded above by any element of S, as all elements of A are less than or equal to every element of S
By the completeness axiom (or the least upper bound property), A has a least upper bound, say g
The completeness axiom states that every non-empty set of real numbers that is bounded above has a least upper bound (LUB)
Proving the Properties of GLB
Prove that g is a lower bound of S
Suppose, for contradiction, that g is not a lower bound of S
Then there exists an element s ∈ S such that s < g
But then s would be an upper bound of A, contradicting the fact that g is the LUB of A
Prove that g is greater than or equal to any other lower bound of S
Let h be any lower bound of S
Then h ∈ A, and since g is the LUB of A, we have h ≤ g
Conclude that g is the GLB of S, and thus the set of real numbers possesses the GLB property
Subsets with Greatest Lower Bounds
Sets Bounded Below
Any non-empty set of real numbers that is bounded below has a GLB
Examples of sets with a GLB:
Closed intervals [a, b], where a and b are real numbers and a ≤ b
Half-open intervals [a, b) or (a, b], where a and b are real numbers and a < b
Sets of the form {x ∈ ℝ : x ≥ a}, where a is a real number
For example, the set {x ∈ ℝ : x ≥ 0} has a GLB of 0
These sets are bounded below because they have a smallest element or a lower bound
Sets Not Bounded Below
Sets that are not bounded below do not have a GLB
Examples of sets without a GLB:
Open intervals (a, ∞), where a is a real number
For example, the set (0, ∞) does not have a GLB because there is no largest real number less than or equal to all elements in the set
The set of all real numbers ℝ
ℝ is not bounded below, so it does not have a GLB
The empty set ∅ does not have a GLB, as it has no lower bounds
By convention, the GLB of the empty set is defined as ∞
Applications of Greatest Lower Bound Property
Proving the Existence of Real Numbers
The GLB property can be used to prove the existence of certain real numbers, such as the existence of sqrt(2)
Define the set S = {x ∈ ℝ : x^2 ≥ 2}
Show that S is non-empty and bounded below
By the GLB property, S has a GLB, say g
Prove that g^2 = 2, and thus sqrt(2) exists
Riemann Integral
The GLB property is used in the definition and properties of the Riemann integral, which is a fundamental concept in real analysis
The Riemann integral of a function f over an interval [a, b] is defined using the infimum of upper sums and the supremum of lower sums
The GLB property ensures the existence of these infimum and supremum values
Archimedean Property
The GLB property can be used to prove the Archimedean property of real numbers
The Archimedean property states that for any positive real numbers x and y, there exists a natural number n such that nx > y
The proof involves defining a set S = {nx : n ∈ ℕ} and using the GLB property to show that x is the GLB of S
Optimization Problems
The GLB property is used in optimization problems to find the minimum value of a function over a given domain
For example, to find the minimum value of a continuous function f on a closed interval [a, b], one can define the set S = {f(x) : x ∈ [a, b]} and use the GLB property to show that S has a GLB, which is the minimum value of f
Limits in Real Analysis
The GLB property can be used to prove the existence of limits of sequences and functions in real analysis
For example, to prove the limit of a monotone increasing sequence, one can define a set S containing the terms of the sequence and use the GLB property to show that S has a GLB, which is the limit of the sequence