Maximum - Knowlpedia

A maximum of a subset $A\subseteq\mathbb R$ is an element $m\in A$ such that $x\le m$ for every $x\in A$ .

Remarks

If a maximum exists, it is unique and equals the supremum of $A$ . Many sets have a supremum but no maximum (for instance, open intervals).

Examples

For $A=[0,1]$ , the maximum is $1$ .
For $A=\{2,5,3\}$ , the maximum is $5$ .

Finite sets

Every nonempty finite set of real numbers has a maximum, by induction using the larger of two numbers. The empty set has no maximum under this definition.