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Greatest Lower Bound of a Set

Greatest Lower Bound of a Set
GLB

The greatest of all lower bounds of a set of numbers. For example, the greatest lower bound of (5, 7) is 5. The greatest lower bound of the interval [5, 7] is also 5.

 

 

See also

Least upper bound, interval notation

Key Formula

GLB(S)=inf⁡(S)=max⁡{ m∈R:m≤x for all x∈S }\text{GLB}(S) = \inf(S) = \max\{\, m \in \mathbb{R} : m \leq x \text{ for all } x \in S \,\}
Where:
  • SS = The set of real numbers whose greatest lower bound you seek
  • mm = A lower bound of S — any number less than or equal to every element of S
  • xx = An arbitrary element of S
  • inf⁡(S)\inf(S) = The infimum (greatest lower bound) of S

Worked Example

Problem: Find the greatest lower bound of the set S = {3, 5, 8, 12}.
Step 1: Identify the lower bounds of S. A lower bound is any number m such that m ≤ x for every x in S. Since the smallest element is 3, any number m ≤ 3 is a lower bound. For instance, 1, 2, 3, −10 are all lower bounds.
m≤3,  m≤5,  m≤8,  m≤12  ⟹  m≤3m \leq 3,\; m \leq 5,\; m \leq 8,\; m \leq 12 \implies m \leq 3
Step 2: Find the greatest among all lower bounds. The set of lower bounds is (−∞, 3]. The largest value in this set is 3.
GLB(S)=max⁡(−∞,3]=3\text{GLB}(S) = \max(-\infty, 3] = 3
Answer: The greatest lower bound of {3, 5, 8, 12} is 3.

Another Example

Problem: Find the greatest lower bound of the open interval S = (2, 6).
Step 1: List some elements near the bottom of S. The set (2, 6) contains numbers like 2.001, 2.0001, 2.00001, and so on, but it does not contain 2 itself.
Step 2: Check whether 2 is a lower bound. Every element x in (2, 6) satisfies x > 2, so 2 ≤ x for all x in S. Yes, 2 is a lower bound.
2≤x for all x∈(2,6)2 \leq x \text{ for all } x \in (2,6)
Step 3: Check whether any number greater than 2 could also be a lower bound. Pick any candidate m > 2, say m = 2.01. Then 2.005 is in S and 2.005 < 2.01, so m = 2.01 fails to be a lower bound. The same argument works for any m > 2.
Step 4: Conclude. Since 2 is a lower bound and no number greater than 2 is a lower bound, the greatest lower bound is 2.
GLB((2,6))=2\text{GLB}\big((2,6)\big) = 2
Answer: The greatest lower bound of (2, 6) is 2, even though 2 is not itself in the set.

Frequently Asked Questions

Does the greatest lower bound have to be an element of the set?
No. The GLB can lie outside the set. For the open interval (2, 6), the GLB is 2, but 2 is not in the set. When the GLB is an element of the set, it equals the minimum of the set. When it is not, the set has an infimum but no minimum.
What is the difference between the greatest lower bound and the minimum of a set?
The minimum is the smallest element that actually belongs to the set, so a minimum exists only if that smallest element is in the set. The greatest lower bound always exists (for any nonempty set bounded below) and equals the minimum when the minimum exists. If the set has no minimum — like (2, 6) — the GLB still exists but sits outside the set.

vs.

The GLB looks downward — it is the largest value that no element falls below. The LUB looks upward — it is the smallest value that no element exceeds. For a set like [3, 9], the GLB is 3 and the LUB is 9. These two concepts are mirror images of each other.

Why It Matters

The greatest lower bound is central to the completeness property of the real numbers, which guarantees that every nonempty set bounded below has an infimum. This property underpins key results in calculus, such as the existence of limits and the Intermediate Value Theorem. In optimization and computer science, finding the GLB of a feasible set tells you the tightest constraint from below.

Common Mistakes

Mistake: Assuming the GLB must be a member of the set.
Correction: The GLB can lie outside the set. For open intervals like (2, 6), the GLB is 2 even though 2 ∉ (2, 6). Only when the set contains its smallest element does the GLB equal the minimum.
Mistake: Confusing 'lower bound' with 'greatest lower bound.'
Correction: A lower bound is any number at or below every element of the set — there are infinitely many. The greatest lower bound is the single largest one among all those lower bounds.

Related Terms