Supremum

analysis set-theory

Definition

Supremum

The supremum (or least upper bound) of a set $S\subseteq \mathbb{R}$ is the smallest real number that is greater than or equal to every element in $S$. It is denoted by $\sup S$.

Formally, $L=\sup S$ if:

1. $L$ is an upper bound: $\forall x\in S:x\le L$.
2. $L$ is the least such bound: if $M$ is any upper bound of $S$, then $L\le M$.

If the set $S$ is not bounded above, we define $\sup S=\infty$. For the empty set, $\sup \varnothing =-\infty$.

Relation to Maximum

A supremum is called a maximum if it is an element of the set $S$. While every non-empty set that is bounded above has a supremum (see Completeness of the Real Numbers), it does not necessarily have a maximum (e.g., the open interval $(0,1)$ has a supremum of $1$ but no maximum).

Properties

• Uniqueness: If a supremum exists, it is unique.
• Approximation Property: For any $\epsilon >0$, there exists an element $x\in S$ such that $x>(\sup S)-\epsilon$.
• Sequences: The supremum of a sequence $(a_n)$ is the supremum of the set of its values $\{a_n\mid n\in \mathbb{N}\}$.