What Is the Difference Between Supremum and Maximum?

The difference is about whether the bound must be an element of the set. A supremum is the least number that is at least as large as every element; a maximum is an element of the set that is at least as large as every other element.

Thus, a maximum is always a supremum, but a supremum need not be a maximum. The distinction matters especially when a set approaches a boundary that it does not include.

Definitions

Let $A$ be a nonempty subset of the real numbers that is bounded above. A number $u$ is an upper bound for $A$ if $x \leq u$ for every $x \in A$. The supremum, written $\sup A$, is the least of all upper bounds. Equivalently, it satisfies two conditions:

1. $x \leq \sup A$ for every $x \in A$.
2. If $v$ is any upper bound for $A$, then $\sup A \leq v$.

A number $m$ is the maximum of $A$, written $\max A$, if $m \in A$ and $x \leq m$ for every $x \in A$. The membership requirement is the key difference.

In $\mathbb{R}$, every nonempty set that is bounded above has a supremum, by the completeness property of the real numbers. It does not necessarily have a maximum.

When the maximum exists

If $m = \max A$, then $m$ is an upper bound for $A$. It is also the least upper bound: any upper bound must be at least as large as the element $m$ itself. Therefore,

$$\max A = \sup A$$

whenever $A$ has a maximum.

The converse does not always hold. A supremum can be a boundary value that the set gets arbitrarily close to but never reaches. If the supremum belongs to the set, however, it is the maximum: $\sup A \in A$ implies $\max A = \sup A$.

Worked examples

An open interval

Take $A = (0,1)$, the set of real numbers strictly between 0 and 1. Every element of $A$ is less than 1, so 1 is an upper bound. No smaller number is an upper bound: if $u < 1$, there is an element of $(0,1)$ greater than $u$. Thus,

$$\sup A = 1.$$

But $1 \notin A$, so there is no maximum. In particular, 0.999 is not a maximum: there are elements of the interval larger than 0.999.

A closed interval

For $B = [0,1]$, the endpoint 1 belongs to the set and is at least as large as every element. Therefore,

$$\max B = \sup B = 1.$$

The only change from the previous example is that the endpoint is included.

A sequence of reciprocals

Let $C = \{1/n : n \in \mathbb{N},\ n \geq 1\}$. Its elements begin $1, 1/2, 1/3, 1/4, \ldots$. The largest element is 1, which belongs to the set. Hence,

$$\max C = \sup C = 1.$$

By contrast, if the set starts at $n=2$, then $D = \{1/n : n \in \mathbb{N},\ n \geq 2\}$ has maximum and supremum both equal to $1/2$. The sequences tend to 0, but 0 is neither the supremum nor the maximum of either set; their maxima are the first terms, 1 and $1/2$.

A practical test for a maximum

For a nonempty, bounded-above set $A$, first find or identify $s = \sup A$. Then check whether $s \in A$:

• If $s \in A$, then $s = \max A$.
• If $s \notin A$, then $A$ has no maximum.

For example, for $A = (0,1)$, the supremum is 1 but 1 is excluded, so no maximum exists. For $B = [0,1]$, the supremum is again 1 and is included, so it is the maximum.

Common mistakes

• Treating any upper bound as the supremum. For $(0,1)$, 2 is an upper bound, but it is not the least upper bound; the supremum is 1.
• Assuming a supremum must be in the set. The definition requires it to be the least upper bound, not a member. The set $(0,1)$ is a standard counterexample.
• Calling an excluded endpoint a maximum. A maximum must be an element of the set. If the interval is open at its upper endpoint, that endpoint cannot be the maximum.
• Assuming every set has a finite supremum or a maximum. A set that is unbounded above, such as the positive integers, has no finite real supremum and no maximum. The empty set also has no maximum; its supremum depends on the convention being used, so the definitions above assume a nonempty set.

When you compare the terms, ask two questions: is the number an upper bound, and does it belong to the set? The least upper bound answers the first question and the maximum must answer both.