Unbounded set

Definition of an unbounded set

A set $ A $ is said to be unbounded if its infimum and supremum are minus infinity and plus infinity, respectively. $$ inf(A)=-\infty $$ $$ sup(A)=+\infty $$

A set is said to be unbounded below if its infimum is equal to minus infinity (-∞).
A set is said to be unbounded above if its supremum is equal to plus infinity (+∞).
A set is called unbounded if both its infimum and supremum are infinite.

More generally, a set $ A \subset \mathbb{R} $ is unbounded below if, for every real number $ M $, there exists at least one element of the set that is strictly smaller than $ M $. Equivalently, the set admits no finite lower bound.

$$ \forall \ M \in \mathbb{R} \ \exists \ x \in A \mid x < M $$

Similarly, a set $ A \subset \mathbb{R} $ is unbounded above if, for every real number $ M $, there exists at least one element of the set that is strictly greater than $ M $. In this case as well, the set admits no finite upper bound.

$$ \forall \ M \in \mathbb{R} \ \exists \ x \in A \mid x > M $$

If a set is unbounded both below and above, it is simply referred to as an unbounded set, without any further qualification.

Note. A set is said to be bounded if it is bounded both below and above, that is, if there exist two real numbers $ m $ and $ M $ such that $ m \le x \le M \quad \forall x \in A $.

A practical example

Example 1

The set of real numbers R is unbounded.

$$ inf(R) = -\infty $$

$$ sup(R) = +\infty $$

In this case, both the infimum and the supremum are infinite.

No matter which real number is chosen, it is always possible to find another real number that is larger and another that is smaller.

Example 2

The set of natural numbers N is unbounded above.

$$ inf(N) = 0 $$

$$ sup(N) = +\infty $$

Here, only the supremum is infinite (+∞), while the infimum (0) is a finite real number.

In other words, given any natural number, one can always find another natural number that is greater.

Example 3

The set of negative real numbers R^- is unbounded below.

$$ inf(R^-) = -\infty $$

$$ sup(R^-) = 0 $$

In this case, only the infimum is infinite (-∞), whereas the supremum (0) is a finite real number.

Equivalently, given any negative real number, it is always possible to find another one that is smaller.

And so on.