Heine-Borel Theorem in $\mathbb{R}^n$

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 $A\subseteq\mathbb{R}^n$. Then the following are equivalent.

1. $A$ is compact in $(\mathbb{R}^n,\mathcal{T}_{d_E})$.

2. $A$ is closed in $(\mathbb{R}^n,\mathcal{T}_{d_E})$ and bounded in the metric space $(\mathbb{R}^n,d_E)$.