Bounded set

A subset $S$ is called bounded in two common contexts:

• In an ordered set $(X,\le)$, a subset $S\subseteq X$ is bounded if it is bounded above and bounded below , i.e. if there exist $\ell,u\in X$ such that

  $$\forall s\in S,\ \ell\le s\le u.$$

• In a metric space $(X,d)$, a subset $S\subseteq X$ is bounded if there exist $x_0\in X$ and $M\in[0,\infty)$ such that

  $$\forall x\in S,\ d(x,x_0)\le M.$$

In $\mathbb{R}$ with its usual metric $d(x,y)=|x-y|$, these two notions agree. In general metric spaces there is no order, so the metric definition is the relevant one.

Examples:

• In $\mathbb{R}$, $S=[-2,5]$ is bounded: $-2\le s\le 5$ for all $s\in S$.
• In $(\mathbb{R}^2,\|\cdot\|_2)$, the unit circle $S=\{x\in\mathbb{R}^2:\|x\|_2=1\}$ is bounded (take $x_0=0$, $M=1$).
• In $\mathbb{R}$, the set $\mathbb{N}$ is not bounded (neither above, nor in the metric sense).