Least Upper Bounds
Definition
Suppose $(A, \leq)$ is a partially ordered set.
An upper bound for $B \subset A$ is an element $a \in A$ so that $b \leq a$ for all $b \in B$. A set is bounded from above if it has a least upper bound. A least upper bound for $B$ is an element $c \in A$ so that $c$ is an upper bound and $c < a$ for all other upper bounds $a$.
If there is a least upper bound it is
unique.1
We call the unique least upper bound of a set
(if it exists) the
supremum.
Notation
We denote the supremum of a set $B \subset A$ by $\sup A$.
- Proof in future editions. ↩︎