Let $X$ be a subset of $\R$ We say that $X$ is bounded when there exists a real number $M$ such that $|x|<M$ for all $x\in X$ When $X$ is an interval, we speak of a bounded interval.

This can be generalized first to $\R^n$ We say that $X\subseteq \R^n$ is bounded if there is a real number $M$ such that $\Vert x\Vert<M$ for all $x\in X$ and $\Vert\cdot\Vert$ is the Euclidean distance between $x$ and $y$

This condition is equivalent to the statement: There is a real number $T$ such that $\Vert x-y\Vert<T$ for all $x,y\in X$

A further generalization to any metric space $V$ says that $X\subseteq V$ is bounded when there is a real number $M$ such that $d(x,y)<M$ for all $x,y\in X$ where $d$ is the metric on $V$