The Borel $\sigma$-Algebras of Euclidean Space and of the Euclidean Metric Coincide

Let $n\ge1$ be a natural number and let $d$ be the Euclidean distance on Euclidean space $\mathbb{R}^n$, which is a metric.

Then the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n,d)$ of the metric space $(\mathbb{R}^n,d)$ and the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ of Euclidean space are the same $\sigma$-algebra on $\mathbb{R}^n$.