Let $n$ be a positive integer, and let $d: \mathbb{R}^n \times \mathbb{R}^n \to [0,\infty)$ be the [Euclidean metric](/page/Euclidean%20Metric) defined by $d(x,y)=|x-y|$. If $A \subset \mathbb{R}^n$ is bounded in the [metric space](/page/Metric%20Space) $(\mathbb{R}^n,d)$, then $A$ is [totally bounded](/page/Totally%20Bounded) in $(\mathbb{R}^n,d)$.