Euclidean metric

Example

Let $n \ge 1$ be an integer. For points $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let $$d_2(x, y) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}.$$ Then $d_2$ is a metric on $\mathbb{R}^n$, the Euclidean metric.