What Is an Example of an Unbounded Set?

A standard example of an unbounded set is the set of natural numbers, $\mathbb{N}$. No matter how large a real number you choose, there is a natural number larger than it. The integers, $\mathbb{Z}$, are another example; they extend without bound in both the positive and negative directions.

What does bounded mean?

For a subset $A$ of the real numbers, the usual definition is that $A$ is bounded if there is some finite real number $M \ge 0$ such that

$$|x| \le M \quad\text{for every }x\in A.$$

In other words, all elements of the set must fit inside the interval $[-M,M]$. If no such $M$ exists, the set is unbounded.

For subsets of $\mathbb{R}$, you can also describe boundedness using upper and lower bounds: a set is bounded if it has both a finite upper bound and a finite lower bound. A set can fail to be bounded in just one direction. For instance, $\mathbb{N}$ is bounded below (under the convention $\mathbb{N}=\{1,2,3,\ldots\}$), but not bounded above.

Worked example: the natural numbers

Take $A=\mathbb{N}=\{1,2,3,\ldots\}$. To show that $A$ is unbounded, assume someone proposes a real number $B$ as an upper bound. The Archimedean property of the real numbers says that there is a natural number larger than any given real number. Thus there is an $n\in\mathbb{N}$ with $n>B$.

So $B$ cannot be an upper bound for $\mathbb{N}$. Since this applies to every proposed finite $B$, the natural numbers have no finite upper bound and are unbounded. Equivalently, given any proposed absolute bound $M$, choose $n>M$; then $|n|>M$.

The same reasoning shows that $\{n^2:n\in\mathbb{N}\}=\{1,4,9,16,\ldots\}$ is unbounded. Its elements eventually exceed any fixed real number, even though they do not include every natural number.

Other examples and one-sided cases

• $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ is unbounded above and below.
• The interval $[0,\infty)$ is unbounded above but bounded below.
• The interval $(-\infty,5]$ is unbounded below but bounded above.
• The set $\{1/n:n\in\mathbb{N}\}$ is bounded: every element lies between $0$ and $1$.

These examples distinguish a set being infinite from being unbounded. An infinite set can be bounded, as $\{1/n:n\in\mathbb{N}\}$ shows. Conversely, unboundedness is about how far the elements extend, not how many elements the set contains.

If the question is about a sequence

A sequence is bounded when its set of values is bounded. For example, the sequence $a_n=n$ is unbounded because its values are the natural numbers. By contrast, $a_n=1/n$ is bounded, since $0<a_n\le 1$ for every positive integer $n$.

A convergent real sequence is always bounded, but an arbitrary sequence need not be. When a textbook asks for an unbounded set in a chapter on sequences or series, the range of a simple sequence such as $a_n=n$ is often an appropriate example.

Common mistakes

• Confusing infinite with unbounded: Infinitely many points can all lie inside a fixed interval. The sequence values $1,1/2,1/3,\ldots$ are one example.
• Checking only one side: $[0,\infty)$ is unbounded even though it has a lower bound. A set needs bounds in both directions to be bounded.
• Using an informal claim without the quantifier: To prove unboundedness, show that for every proposed bound $M$, some element of the set exceeds it in absolute value. For $\mathbb{N}$, choose $n>M$.
• Forgetting the ambient space: Here, boundedness is being considered in the usual distance on $\mathbb{R}$. In a general metric space, boundedness means having finite diameter: distances between pairs of points in the set have a common finite upper bound.