# Statement(s) > [!NOTE] Statement 1 (Euclidean Metric on $\mathbb{R}^n$ is a Metric) > Let $d$ denote the [[Euclidean Metric on Real n-Space|standard Euclidean metric on the real n-space]]. Then $d$ is a [[Metrics|metric]] on $\mathbb{R}^n.$ # Proof(s) **Proof of statement 1:** Necessary to show that $d$ is non-degenerate, symmetric and satisfies the triangle inequality. Let $\mathbf{x}=(x_{1},x_{2},\dots,x_{n}),\mathbf{y}=(y_{1},y_{2},\dots,y_{n})\in \mathbb{R}^{n}.$ **Non-degenerate**: Since $x_{i}$ and $y_{i}$ are real $(x_{i}-y_{i})^{2}=0\iff x_{i}=y_{i}.$ Hence $d(\mathbf{x},\mathbf{y})=0$ iff $\mathbf{x}=\mathbf{y}.$ **Symmetric**: Since $(x_{i}-y_{i})^2 =(y_{i}-x_{i})^2,$ $d(\mathbf{x},\mathbf{y})=d(\mathbf{y},\mathbf{x}).$ **Triangle inequality**: See [[Euclidean spaces are normed spaces]]. $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography