Let $\mathbb R$ denote the [real numbers](/page/Real%20Numbers), let $(X,d)$ be a [metric space](/page/Metric%20Space), and let $A \subset X$. For $x\in X$ and $r>0$, write $B_d(x,r):=\{y\in X:d(x,y)<r\}$, and let $\mathbb N=\{1,2,3,\dots\}$. If $A$ is totally bounded, meaning that for every $\varepsilon>0$ there exist $m\in\mathbb N$ and points $x_1,\dots,x_m\in X$ such that \begin{align*} A\subset \bigcup_{i=1}^{m} B_d(x_i,\varepsilon), \end{align*} then $A$ is bounded, meaning that either $A=\varnothing$ or there exist $x_0\in X$ and $R>0$ such that $A\subset B_d(x_0,R)$.