Definition 5

Let $E$ be a preordered set and let $X$ be a subset of $E$. Any element $x\in E$ such that $x\leq y$ (resp. $x\geq y$) for all $y\in X$ is called a lower (resp. upper) bound of $X$ in $E$.

Every upper bound of $X$ is a lower bound of $X$ with respect to the opposite ordering, and vice versa.

¶ If $x$ is a lower bound of $X$, every element $z\leq x$ is also a lower bound of $X$. A lower bound of $X$ is also a lower bound of every subset of $X$. An ordered set $X$ has a least element if and only if there exists a lower bound of $X$ which belongs to $X$.

¶ The set of lower bounds of a subset $X$ of a preordered set $E$ may be empty : this is the case when $X=E$ and $E$ is an ordered set which has no least element.

¶ A subset $X$ of $E$ whose set of lower (resp. upper) bounds is not empty is said to be bounded below (resp. bounded above). A subset which is bounded both below and above is said to be bounded. If $X$ is bounded below (resp. bounded above, bounded), the same is true of every subset of $X$.

Every subset consisting of a single element is bounded. But a subset consisting of two elements need not be bounded either above or below (no. 10).

Let $E$ be a preordered set and let $f$ be a mapping of an arbitrary set $A$ into $E$. The mapping $f$ is said (by abuse of language) to be bounded below (resp. bounded above, bounded) if the set $f(A)$ is bounded below (resp. bounded above, bounded) in $E$.