Supremum

A supremum of a nonempty set $A\subseteq\mathbb R$ that is bounded above is a real number $s\in\mathbb R$ such that:

1. $s$ is an upper bound of $A$ (i.e., $x\le s$ for all $x\in A$), and
2. for every upper bound $u$ of $A$, one has $s\le u$.

The supremum is the “least upper bound” and may exist even when $A$ has no maximum . The completeness axiom asserts that every nonempty bounded-above set of real numbers has a supremum.

Examples:

• If $A=(0,1)$, then $\sup A=1$.
• If $A=\{1-\tfrac1n : n\in\mathbb N\}$, then $\sup A=1$ (even though $1\notin A$).