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

Write $C=\operatorname{cl}_{\mathbb{R}^n}(A)$ for the closure of $A$ in the topological space $(\mathbb{R}^n,\mathcal{T}_{d_E})$.

By claim 2 of The Closure is the Smallest Closed Superset, applied to the topological space $(\mathbb{R}^n,\mathcal{T}_{d_E})$ and the subset $A$, the set $C$ is closed in $(\mathbb{R}^n,\mathcal{T}_{d_E})$.

By The Closure of a Bounded Subset of a Metric Space is Bounded, applied to the metric space $(\mathbb{R}^n,d_E)$ and the subset $A$, which is bounded in $(\mathbb{R}^n,d_E)$ by hypothesis, the set $C$ is bounded in $(\mathbb{R}^n,d_E)$.

Thus $C$ satisfies condition 2 of Heine-Borel Theorem in $\mathbb{R}^n$, and that theorem gives condition 1, namely that $C$ is compact in $(\mathbb{R}^n,\mathcal{T}_{d_E})$.