Extrema of a set

What extrema are

Every nonempty subset of the real numbers admits both a lower extremum and an upper extremum, possibly infinite.

• For bounded sets, the extrema are finite real numbers.
• For unbounded sets, the extrema are the symbols plus or minus infinity (±∞).

A set may be bounded above, with a finite upper extremum, and unbounded below, with an infinite lower extremum, or vice versa.

Note. The extrema of a set may or may not belong to the set itself.

To gain a solid understanding of these ideas, it is helpful to examine lower and upper extrema through explicit and concrete examples.

Lower extremum

Let \( A \subseteq \mathbb{R} \) be a set and let B be a nonempty subset $$ B \subseteq A $$ The lower extremum, or infimum, of B is the real number \( a \) such that $$ a \le b \:\:\: \forall \, b \in B $$ and such that any real number strictly greater than \( a \) fails to be a lower bound of B. $$ inf(B) = a $$

Equivalently, the lower extremum is the greatest of all lower bounds of the set B.

The lower extremum of a set may or may not belong to the set itself. If it does belong to the set, it is called the minimum.

For example, the lower extremum of B is the number 3.

For bounded sets, the lower extremum is a finite real number and coincides with the maximum of the set of all lower bounds (minorants). 

For sets that are unbounded below, the lower extremum is minus infinity (-∞).

Example 1

The set A is the set of all real numbers \( \mathbb{R} \).

The set B consists of seven real numbers and is a subset of A.

$$ B = \{ 3, 4, 7, 5, 8, 9, 6 \} $$

The lower extremum of the set B is the number three.

$$ inf(B) = 3 \le b \:\:\: \forall \, b \in B $$

because the real number 3 is less than or equal to every element of the set B.

Note. In this case, the lower extremum \( inf(B)=3 \) belongs to the set B. This need not hold in general.

Example 2

The set A is the set of all real numbers \( \mathbb{R} \).

The set B is the set of positive real numbers \( \mathbb{R}^+ \) and is a subset of A.

The lower extremum of the set B is zero.

$$ inf(B) \le b \:\:\: \forall \, b \in B $$

It is the greatest of all lower bounds of the set \( \mathbb{R}^+ \).

Note. In this case, the lower extremum \( inf(B)=0 \) does not belong to the set B \( (\mathbb{R}^+) \), because between zero and any positive real number there exist infinitely many other points.

Upper extremum

Let \( A \subseteq \mathbb{R} \) be a set and let B be a nonempty subset $$ B \subseteq A $$ The upper extremum, or supremum, of B is the real number \( a \) such that $$ a \ge b \:\:\: \forall \, b \in B $$ and such that any real number strictly smaller than \( a \) fails to be an upper bound of B. $$ sup(B) = a $$

Equivalently, the upper extremum is the smallest of all upper bounds of the set B.

The upper extremum may or may not belong to the set itself. If it does belong to the set, it is called the maximum.

For example, the upper extremum of B is 4.

For bounded sets, the upper extremum is a finite real number and coincides with the minimum of the set of all upper bounds (majorants).

For sets that are unbounded above, the upper extremum is plus infinity (+∞).

Example 1

The set A is the set of all real numbers \( \mathbb{R} \).

The set B consists of seven real numbers and is a subset of A.

$$ B = \{ 3, 4, 7, 5, 8, 9, 6 \} $$

The upper extremum of the set B is the number nine.

$$ sup(B) = 9 \ge b \:\:\: \forall \, b \in B $$

because the real number 9 is greater than or equal to every element of the set B.

Note. In this case, the upper extremum \( sup(B)=9 \) belongs to the set B. This need not hold in general.

Example 2

The set A is the set of all real numbers \( \mathbb{R} \).

The set B is the set of negative real numbers \( \mathbb{R}^- \) and is a subset of A.

The lower extremum of the set B is minus infinity.

$$ inf(B) = -\infty \le b \:\:\: \forall \, b \in B $$

Note. In this case, the lower extremum \( inf(B)=-\infty \) belongs to the extended real line but not to the set B \( (\mathbb{R}^-) \), which is unbounded below. By contrast, the upper extremum \( sup(B)=0 \) does not belong to the set B, because zero is not a negative real number.

Extrema of a real set

Equivalently, the lower and upper extrema of a real set can be characterized as follows.

Lower extremum of a real set

Let \( E \subset \mathbb{R} \) be a set that is bounded below. The lower extremum of \( E \) is the real number \( m \) such that:

• for every \( x \in E \), one has \( x \ge m \);
• for every \( \varepsilon > 0 \), there exists at least one element \( x \in E \) such that \( x < m + \varepsilon \).

In this case, one writes \( m = \inf E \).

This formulation expresses the fact that \( m \) is the greatest of all lower bounds of \( E \).

Upper extremum of a real set

Let \( E \subset \mathbb{R} \) be a set that is bounded above. The upper extremum of \( E \) is the real number \( M \) such that:

• for every \( x \in E \), one has \( x \le M \);
• for every \( \varepsilon > 0 \), there exists at least one element \( x \in E \) such that \( x > M - \varepsilon \).

In this case, one writes \( M = \sup E \).

This formulation expresses the fact that \( M \) is the smallest of all upper bounds of \( E \).

Notes

Additional remarks and observations

• Every nonempty subset of the real numbers admits both a lower extremum and an upper extremum
  In particular, if a set is unbounded above, its upper extremum is \( \sup A = +\infty \). Likewise, if a set is unbounded below, its lower extremum is \( \inf A = -\infty \). If a set is bounded above, it has a finite and unique upper extremum. Similarly, every set that is bounded below has a finite and unique lower extremum. In this sense, the notion of an extremum is always well defined when working within the framework of the extended real numbers.

  By "extended real numbers" we mean the real line \( \mathbb{R} \) augmented with the symbols \( -\infty \) and \( +\infty \). These symbols are not real numbers, but they are introduced to ensure that lower and upper extrema are always defined.

And so on.