bounded set

A set is bounded if all its elements are within a certain distance of each other.

Formally,
a subset $S$ of a metric space $(M,d)$ is bounded if there exists an $r>0$ such that
for all $s,t\in S$, then $d(s,t)<r$.

A subset of real numbers $S\subseteq \mathbb{R}$ is bounded from above or bounded from below if there exists an upper or lower bound, respectively.

The "boundaries" of bounded set are its supremum and infimum.

If the supremum and infimum are contained in the set, they are the greatest and least elements, i.e. the maximum and minimum.