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

Let $n\in\mathbb{N}$, and let $A\subseteq\mathbb{R}^n$. Then the following are equivalent.

1. $A$ is compact in $\mathbb{R}^n$, where $\mathbb{R}^n$ is regarded as a topological space through the topology determined by the Euclidean distance.
2. $A$ is closed in $\mathbb{R}^n$ and bounded as a subset of the metric space $(\mathbb{R}^n,d_E)$.