Upper bound

An upper bound of a subset $S$ of an ordered set $(X,\le)$ is an element $u\in X$ such that

Upper bounds formalize the idea that a set lies entirely to the “left” of some point. The existence and structure of upper bounds is central to completeness and to definitions such as supremum.

Examples:

• In $(\mathbb{R},\le)$, the set $S=(0,1)$ has upper bounds $u=1$, $u=2$, and in fact every $u\ge 1$.
• In $(\mathbb{R},\le)$, the set $S=\{x\in\mathbb{R}: x<0\}$ has upper bounds $u=0$ and every $u\ge 0$.
• In $(\mathbb{Z},\le)$, the set $S=\{n\in\mathbb{Z}: n\le 5\}$ has upper bound $u=5$ (and any $u\ge 5$).