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Lower Bound of a Set

Lower Bound of a Set

Any number that is less than or equal to all of the elements of a given set. For example, 5 is a lower bound of the interval [8,9]. So are 6, 7, and 8.

 

 

 

See also

Greatest lower bound, upper bound, interval notation

Key Formula

L is a lower bound of S  ⟺  L≤x for all x∈SL \text{ is a lower bound of } S \iff L \leq x \text{ for all } x \in S
Where:
  • LL = A candidate lower bound (a real number)
  • SS = The given set of numbers
  • xx = Any element belonging to the set S

Worked Example

Problem: Find all integer lower bounds of the set S = {3, 7, 10, 15}.
Step 1: Identify the smallest element of S. The elements are 3, 7, 10, and 15, so the smallest is 3.
min⁡(S)=3\min(S) = 3
Step 2: A lower bound must be less than or equal to every element. Since 3 is the smallest, any lower bound L must satisfy L ≤ 3.
L≤3L \leq 3
Step 3: Check a few integers: L = 3 satisfies 3 ≤ 3, 3 ≤ 7, 3 ≤ 10, 3 ≤ 15 — yes. L = 2 satisfies 2 ≤ 3, 2 ≤ 7, 2 ≤ 10, 2 ≤ 15 — yes. L = 4 fails because 4 > 3 — no.
3≤3✓,2≤3✓,4≤3×3 \leq 3 \checkmark, \quad 2 \leq 3 \checkmark, \quad 4 \leq 3 \times
Step 4: The integer lower bounds are ..., −2, −1, 0, 1, 2, 3. In fact, every real number less than or equal to 3 is a lower bound. The greatest lower bound is 3 itself.
Integer lower bounds: {…,−2,−1,0,1,2,3}\text{Integer lower bounds: } \{\ldots, -2, -1, 0, 1, 2, 3\}
Answer: Every integer less than or equal to 3 is a lower bound of S = {3, 7, 10, 15}. The greatest lower bound is 3.

Another Example

Problem: Is 2 a lower bound of the interval (2, 5]?
Step 1: The interval (2, 5] contains all real numbers x with 2 < x ≤ 5. Note that 2 itself is NOT in the set because the parenthesis means the endpoint is excluded.
(2,5]={x∈R:2<x≤5}(2, 5] = \{x \in \mathbb{R} : 2 < x \leq 5\}
Step 2: Check whether 2 ≤ x for every x in (2, 5]. Since every element x satisfies x > 2, we know 2 < x, which certainly means 2 ≤ x.
x>2  ⟹  2≤xx > 2 \implies 2 \leq x
Step 3: So 2 is a lower bound of (2, 5], even though 2 is not an element of the set. Could any number greater than 2 also be a lower bound? No — for any L > 2, you can find an element of (2, 5] between 2 and L, so L would fail.
Answer: Yes, 2 is a lower bound of (2, 5]. It is also the greatest lower bound, despite not belonging to the set.

Frequently Asked Questions

Does a lower bound have to be in the set?
No. A lower bound can be any number — it does not need to belong to the set. For example, 0 is a lower bound of {5, 6, 7}, even though 0 is not in the set. When the greatest lower bound does belong to the set, it equals the minimum of the set.
Can a set have more than one lower bound?
Yes. If L is a lower bound, then every number less than L is also a lower bound. So a set that has any lower bound at all has infinitely many. The most important one is the greatest lower bound (infimum), which is the largest of all the lower bounds.

Lower Bound vs. Upper Bound

A lower bound L satisfies L ≤ x for every element x in the set, while an upper bound U satisfies U ≥ x for every element x. They mirror each other: one caps the set from below, the other from above. For the set [3, 8], any number ≤ 3 is a lower bound and any number ≥ 8 is an upper bound.

Why It Matters

Lower bounds help define whether a set is "bounded below," which is essential in calculus and analysis. The Completeness Property of the real numbers guarantees that every nonempty set with a lower bound has a greatest lower bound (infimum). This property underpins key theorems about limits, convergence of sequences, and the existence of integrals.

Common Mistakes

Mistake: Thinking the lower bound must be the smallest element of the set.
Correction: A lower bound is any number at or below every element of the set — there are usually infinitely many. The smallest element (if it exists) is called the minimum, and it equals the greatest lower bound in that case.
Mistake: Believing that a lower bound must be a member of the set.
Correction: A lower bound is defined by a comparison to all elements; it need not belong to the set itself. For instance, the set (0, 1) has greatest lower bound 0, but 0 is not in the set.

Related Terms