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Bounded Set — Definition, Formula & Examples

A bounded set is a set that can be contained within some finite interval or region — meaning its elements do not stretch out to infinity in any direction. Equivalently, a set of real numbers is bounded if it has both an upper bound and a lower bound.

A set S⊆RS \subseteq \mathbb{R} is bounded if there exists a real number M>0M > 0 such that ∣x∣≤M|x| \leq M for every x∈Sx \in S. This is equivalent to requiring that SS has both a finite upper bound and a finite lower bound.

Key Formula

∃ M>0 such that ∣x∣≤M    ∀ x∈S\exists\, M > 0 \text{ such that } |x| \leq M \;\; \forall\, x \in S
Where:
  • SS = The set being tested for boundedness
  • MM = A positive real number that serves as the bound
  • xx = An arbitrary element of the set S

How It Works

To check whether a set of real numbers is bounded, look for a single number MM such that every element of the set lies between −M-M and MM. If you can find such an MM, the set is bounded. If the set extends arbitrarily far in the positive or negative direction (or both), no such MM exists and the set is unbounded. In metric spaces more generally, a set is bounded if it fits inside some ball of finite radius.

Worked Example

Problem: Determine whether the set S={1,  3,  5,  7,  9}S = \{1,\; 3,\; 5,\; 7,\; 9\} is bounded.
Step 1: Identify the smallest and largest elements of S.
min⁡(S)=1,max⁡(S)=9\min(S) = 1, \quad \max(S) = 9
Step 2: Choose an M such that every element satisfies ∣x∣≤M|x| \leq M. Since all elements lie between 1 and 9, pick M=9M = 9.
∣x∣≤9    for all x∈S|x| \leq 9 \;\; \text{for all } x \in S
Step 3: Since such an M exists, the set is bounded.
Answer: SS is bounded, with M=9M = 9 (or any larger value) serving as the bound.

Why It Matters

Boundedness is a prerequisite for many core theorems in analysis and calculus. The Bolzano–Weierstrass theorem guarantees that every bounded sequence in Rn\mathbb{R}^n has a convergent subsequence. The Extreme Value Theorem states that a continuous function on a closed, bounded interval attains its maximum and minimum — a result used constantly in optimization problems.

Common Mistakes

Mistake: Confusing bounded with closed. Students sometimes assume every bounded set is closed (or vice versa).
Correction: Boundedness and closedness are independent properties. The open interval (0,1)(0, 1) is bounded but not closed. The set [0,∞)[0, \infty) is closed but not bounded.