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

By A Compact Subset of a Metric Space is Totally Bounded, applied to the metric space $(\mathbb{R}^n,d_E)$ and the subset $A$, compactness of $A$ in $(\mathbb{R}^n,\mathcal{T}_{d_E})$ gives that $A$ is totally bounded in $(\mathbb{R}^n,d_E)$.

The underlying set $\mathbb{R}^n$ is nonempty, since it contains the origin $0_{\mathbb{R}^n}$. Hence A Totally Bounded Subset of a Nonempty Metric Space is Bounded applies to $(\mathbb{R}^n,d_E)$ and $A$, and shows that $A$ is bounded in $(\mathbb{R}^n,d_E)$.