Bounds

Boundedness

Definition 1(Definition).

Suppose $S$ is an ordered set, $E\subset S$. If there exists a $\beta \in S$ such that $x\le \beta$ for every $x\in E$, we say that $E$ is bounded above, and call $\beta$ an upper bound of $E$.

Supremum and infimum

Definition 2(Definition).

Suppose $S$ is an ordered set, $E\subset S$, and $E$ is bounded above. Suppose there exists an $\alpha \in S$ with the following properties:

• $\alpha$ is an upper bound of $E$.
• If $\gamma <\alpha$, then $\gamma$ is not an upper bound of $E$.

Then $\alpha$ is called the least upper bound or supremum of $E$.

Analogous definitions exist for bounded below and infimum.
Note that $\sup A$ and $\inf A$ may or may not be in $A$.