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Unbounded Set of Numbers

Unbounded Set of Numbers

A set of numbers that is not bounded. That is, a set that lacks either a lower bound or an upper bound. For example, the sequence 1, 2, 3, 4,... is unbounded.

 

Worked Example

Problem: Determine whether the set S = {2, 4, 8, 16, 32, 64, ...} is bounded or unbounded.
Step 1: Check for a lower bound. Every element of S is at least 2, so 2 is a lower bound. In fact, every element is positive, so 0 also works as a lower bound. The set is bounded below.
x≥2 for all x∈Sx \geq 2 \text{ for all } x \in S
Step 2: Check for an upper bound. We need a single number M such that every element of S is less than or equal to M. The elements are powers of 2: the nth element is 2^n. As n increases, 2^n grows without limit.
2n→∞ as n→∞2^n \to \infty \text{ as } n \to \infty
Step 3: Since for any proposed upper bound M, we can always find a power of 2 that exceeds M, no finite upper bound exists. The set is not bounded above.
For any M>0, there exists n such that 2n>M\text{For any } M > 0, \text{ there exists } n \text{ such that } 2^n > M
Answer: S = {2, 4, 8, 16, 32, 64, ...} is unbounded because it has no finite upper bound, even though it is bounded below by 2.

Another Example

Problem: Determine whether the set T = {..., -27, -9, -3, -1} is bounded or unbounded.
Step 1: Check for an upper bound. Every element is at most -1, so -1 is an upper bound. The set is bounded above.
x≤−1 for all x∈Tx \leq -1 \text{ for all } x \in T
Step 2: Check for a lower bound. The elements extend as -1, -3, -9, -27, ... and the pattern continues with -3^n, which becomes more and more negative without limit.
−3n→−∞ as n→∞-3^n \to -\infty \text{ as } n \to \infty
Step 3: No finite lower bound exists because the set contains arbitrarily large negative numbers.
For any L<0, there exists n such that −3n<L\text{For any } L < 0, \text{ there exists } n \text{ such that } -3^n < L
Answer: T is unbounded because it has no finite lower bound, even though it is bounded above by -1.

Frequently Asked Questions

Can a set be unbounded in only one direction?
Yes. A set can be unbounded above (no upper bound) while still being bounded below, or unbounded below (no lower bound) while still being bounded above. For example, {1, 2, 3, 4, ...} is bounded below by 1 but unbounded above. A set is called unbounded if it fails to be bounded in at least one direction.
Is the set of all integers unbounded?
Yes. The integers {..., -3, -2, -1, 0, 1, 2, 3, ...} are unbounded in both directions. There is no finite upper bound because the positive integers grow without limit, and there is no finite lower bound because the negative integers decrease without limit.

Unbounded Set vs. Bounded Set

A bounded set has both a finite upper bound and a finite lower bound — all its elements fit within some fixed interval [a,b][a, b]. An unbounded set fails at least one of these conditions: its elements either grow arbitrarily large, decrease without limit, or both. For instance, {−5,−3,0,2,7}\{-5, -3, 0, 2, 7\} is bounded (it fits within [−5,7][-5, 7]), while {1,2,3,4,...}\{1, 2, 3, 4, ...\} is unbounded because no finite number can serve as an upper bound for all elements.

Why It Matters

Understanding whether a set is bounded or unbounded is essential in calculus and analysis. Many theorems, such as the Extreme Value Theorem, require functions to be defined on bounded, closed intervals. When working with sequences or series, recognizing that partial sums form an unbounded set tells you the series diverges.

Common Mistakes

Mistake: Thinking a set must lack both an upper bound and a lower bound to be unbounded.
Correction: A set is unbounded if it lacks even one of the two bounds. The set {1, 2, 3, ...} is bounded below by 1 but has no upper bound, so it is still unbounded.
Mistake: Confusing an infinite set with an unbounded set.
Correction: A set can be infinite yet bounded. For example, the set of all real numbers between 0 and 1 is infinite (uncountably so), but it is bounded because every element satisfies 0≤x≤10 \leq x \leq 1.

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