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Least Upper Bound of a Set — Definition & Examples

Least Upper Bound of a Set
LUB

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

 

 

See also

Greatest lower bound, interval notation

Key Formula

If S⊆R, then L=sup⁡(S)  ⟺  {x≤Lfor all x∈Sno L′<L is an upper bound of S\text{If } S \subseteq \mathbb{R}, \text{ then } L = \sup(S) \iff \begin{cases} x \le L & \text{for all } x \in S \\ \text{no } L' < L \text{ is an upper bound of } S \end{cases}
Where:
  • SS = A nonempty set of real numbers
  • LL = The least upper bound (supremum) of S
  • xx = Any element of the set S
  • L′L' = Any value strictly less than L

Worked Example

Problem: Find the least upper bound of the set S = {1, 3, 5, 8, 10}.
Step 1: Identify the upper bounds. An upper bound is any number that is greater than or equal to every element in S. Since the largest element is 10, any number ≥ 10 is an upper bound.
Upper bounds: 10,  11,  15,  100,…\text{Upper bounds: } 10,\; 11,\; 15,\; 100, \ldots
Step 2: Find the smallest upper bound. Among all numbers that are ≥ 10, the smallest one is 10 itself.
sup⁡(S)=10\sup(S) = 10
Step 3: Verify: 10 is ≥ every element of S, and no number smaller than 10 (say 9.99) is ≥ every element, since 9.99 < 10 ∈ S.
9.99<10∈S  ⟹  9.99 is not an upper bound9.99 < 10 \in S \implies 9.99 \text{ is not an upper bound}
Answer: The least upper bound (supremum) of {1, 3, 5, 8, 10} is 10.

Another Example

Problem: Find the least upper bound of the open interval S = (2, 6).
Step 1: The set S contains all real numbers strictly between 2 and 6, so 6 is not in S. However, any number ≥ 6 is still an upper bound of S, because every element of S is less than 6.
x<6 for all x∈(2,6)x < 6 \text{ for all } x \in (2,6)
Step 2: Check whether any number smaller than 6 could be an upper bound. Pick any candidate L' < 6. Then choose a number between L' and 6, for instance (L' + 6)/2, which lies in S but is greater than L'. So L' fails as an upper bound.
L′<L′+62<6  ⟹  L′+62∈S and L′+62>L′L' < \frac{L' + 6}{2} < 6 \implies \frac{L'+6}{2} \in S \text{ and } \frac{L'+6}{2} > L'
Step 3: Since 6 is an upper bound and no smaller number qualifies, 6 is the least upper bound — even though 6 is not in the set.
sup⁡((2,6))=6\sup\big((2, 6)\big) = 6
Answer: The least upper bound of (2, 6) is 6, despite 6 not being a member of the set.

Frequently Asked Questions

Can the least upper bound be a member of the set?
Yes, it can be. For a finite set like {2, 4, 6}, the least upper bound is 6, which is in the set. For a closed interval like [3, 9], the LUB is 9, also in the set. But it doesn't have to be — the open interval (3, 9) has LUB 9, which is not in the set.
What is the difference between supremum and maximum?
The maximum of a set is the largest element that actually belongs to the set. The supremum (least upper bound) is the smallest value ≥ every element, and it may or may not belong to the set. If a set has a maximum, then the supremum equals the maximum. But a set can have a supremum without having a maximum — for example, (0, 1) has sup = 1 but no maximum.

Least Upper Bound (Supremum) vs. Greatest Lower Bound (Infimum)

The least upper bound is the smallest value ≥ every element of the set, while the greatest lower bound (infimum) is the largest value ≤ every element. They bound the set from opposite sides. For instance, the set (2, 6) has supremum 6 and infimum 2.

Why It Matters

The least upper bound is a cornerstone of real analysis. The Least Upper Bound Property — the fact that every nonempty set of real numbers bounded above has a supremum in the reals — is what distinguishes the real numbers from the rationals. This property underpins limits, continuity, and the rigorous definition of integrals.

Common Mistakes

Mistake: Assuming the least upper bound must be an element of the set.
Correction: The LUB only needs to be ≥ every element and be the smallest such value. It does not have to belong to the set. For example, sup(0, 1) = 1, but 1 ∉ (0, 1).
Mistake: Confusing "upper bound" with "least upper bound."
Correction: A set can have infinitely many upper bounds. The number 100 is an upper bound of {1, 2, 3}, but the least upper bound is 3. Always look for the smallest upper bound, not just any upper bound.

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