Let $n$ be a positive integer and let $A\subset\mathbb{R}^n$. Equip $\mathbb{R}^n$ with the [Euclidean metric](/page/Euclidean%20Metric) $d(x,y)=|x-y|$, where for $z=(z_1,\dots,z_n)\in\mathbb{R}^n$ one has $|z|=\left(\sum_{i=1}^{n}z_i^2\right)^{1/2}$. Then $A$ is bounded in this metric if and only if there exists $M>0$ such that

\begin{align*} A\subset [-M,M]^n=\{x=(x_1,\dots,x_n)\in\mathbb{R}^n: |x_i|\le M \text{ for every } i\in\{1,\dots,n\}\}. \end{align*}