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Bounded Set of Numbers

Bounded Set of Numbers

A set of numbers with both an upper bound and a lower bound. For example, the interval [3,7) is bounded. So is the set Set notation showing {1, 1/2, 1/3, 1/4, ...}, a bounded set of numbers with decreasing fractions..

 

 

See also

Interval notation, bounded sequence

Key Formula

A set S is bounded if there exist real numbers m and M such that m≤x≤M for all x∈S.\text{A set } S \text{ is bounded if there exist real numbers } m \text{ and } M \text{ such that } m \leq x \leq M \text{ for all } x \in S.
Where:
  • SS = The set of numbers being examined
  • mm = A lower bound of the set
  • MM = An upper bound of the set
  • xx = Any element belonging to the set S

Worked Example

Problem: Determine whether the set S = {−2, 0, 3, 5, 8} is bounded.
Step 1: Find the smallest element in the set to check for a lower bound.
min⁡(S)=−2\min(S) = -2
Step 2: Find the largest element in the set to check for an upper bound.
max⁡(S)=8\max(S) = 8
Step 3: Choose a lower bound m and an upper bound M. Any m ≤ −2 and any M ≥ 8 will work. For instance, pick m = −2 and M = 8.
−2≤x≤8for all x∈S-2 \leq x \leq 8 \quad \text{for all } x \in S
Step 4: Since both bounds exist, the set is bounded.
Answer: The set S = {−2, 0, 3, 5, 8} is bounded, with lower bound −2 and upper bound 8.

Another Example

Problem: Is the set T = {x ∈ ℝ : x > 0} bounded?
Step 1: Check for a lower bound. Every element of T is greater than 0, so m = 0 serves as a lower bound.
0≤xfor all x∈T0 \leq x \quad \text{for all } x \in T
Step 2: Check for an upper bound. For any candidate M, you can always find x = M + 1 in T, which exceeds M. No upper bound exists.
For any M, choose x=M+1>M, and x∈T\text{For any } M, \text{ choose } x = M + 1 > M, \text{ and } x \in T
Step 3: Since the set has a lower bound but no upper bound, it is not bounded — it is only bounded below.
Answer: The set T = {x ∈ ℝ : x > 0} is NOT bounded. It is bounded below (by 0) but not bounded above.

Frequently Asked Questions

What is the difference between a bounded set and a bounded sequence?
A bounded set is any collection of numbers that fits between a lower bound and an upper bound. A bounded sequence is specifically an ordered list of numbers (indexed by positive integers) whose terms all lie between some lower and upper bound. Every bounded sequence forms a bounded set, but a bounded set is not necessarily a sequence because it may lack an ordering or indexing.
Can an infinite set be bounded?
Yes. A set can have infinitely many elements and still be bounded. For example, the interval [0, 1] contains infinitely many real numbers, yet every element satisfies 0 ≤ x ≤ 1, so it is bounded.

Bounded set vs. Unbounded set

A bounded set has both an upper bound and a lower bound, meaning all its elements are contained within a finite range. An unbounded set is missing at least one of these bounds — it extends to infinity in at least one direction. For instance, [−5, 10] is bounded, while (−∞, 10] is unbounded because it has no lower bound.

Why It Matters

Boundedness is a key condition in many theorems across calculus and analysis. For example, the Extreme Value Theorem guarantees that a continuous function on a closed, bounded interval attains a maximum and minimum value. Understanding bounded sets also prepares you for topics like convergence of sequences and optimization problems.

Common Mistakes

Mistake: Thinking a set is bounded just because it has a lower bound (or just an upper bound).
Correction: A set must have BOTH an upper bound and a lower bound to be called bounded. A set with only one is called 'bounded above' or 'bounded below,' not simply 'bounded.'
Mistake: Confusing the bound with an element that must belong to the set.
Correction: An upper or lower bound does not need to be an element of the set itself. For the open interval (0, 1), the number 1 is an upper bound even though 1 is not in the set.

Related Terms