Closed Boxes in $\mathbb{R}^n$ Are Compact

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $n\in\mathbb{N}$, and let $d_{E}$ be the Euclidean distance on $\mathbb{R}^{n}$, a metric by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence (i), with components as in Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii). Let $a,b\in\mathbb{R}^{n}$ with $a_{k}\le b_{k}$ for every $k\in[n]$, and let

$$Q=\{x\in\mathbb{R}^{n}:a_{k}\le x_{k}\le b_{k}\text{ for every }k\in[n]\}.$$

(i) $Q$ is compact in $(\mathbb{R}^{n},\mathcal{T}_{d_{E}})$.