The Closure of a Bounded Subset of $\mathbb{R}^n$ is Compact

Let $n$ be a natural number, let $d_E$ be the Euclidean distance on Euclidean space $\mathbb{R}^n$, which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$, and let $\mathcal{T}_{d_E}$ be the collection of subsets of $\mathbb{R}^n$ that are open in $(\mathbb{R}^n,d_E)$, which is a topology on $\mathbb{R}^n$ by Metric Open Sets Form a Topology. Let $A\subseteq\mathbb{R}^n$ be bounded in $(\mathbb{R}^n,d_E)$.

Then the closure $\operatorname{cl}_{\mathbb{R}^n}(A)$ of $A$ in the topological space $(\mathbb{R}^n,\mathcal{T}_{d_E})$ is compact in $(\mathbb{R}^n,\mathcal{T}_{d_E})$.