Upper bound

Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$. An element $u \in P$ is an upper bound of $S$ when $s \le u$ for each $s \in S$. The set $S$ is bounded above when it has an upper bound.

Remarks

An upper bound need not be in $S$. The number $1$ is an upper bound of $[0, 1)$ in $\mathbb{R}$, and so is $5$. A lower bound is defined in the same way with $u \le s$. The least upper bound of $S$, when it exists, is the supremum of $S$.

Explanations

L1 human

An upper bound is a ceiling for the set: no element of the set is above it. The set $\{1, 1/2, 1/3, \dots\}$ has upper bounds $1$, $2$, $100$ and many more.