Unit n-Sphere under Euclidean Metric is Metric Subspace of Euclidean Real Vector Space

Theorem

Let $\Bbb S^n$ be the unit $n$-sphere.

Let $d_S: \Bbb S^n \times \Bbb S^n \to \R$ be the real-valued function defined as:

$\ds \forall x, y \in \Bbb S^n: \map {d_S} {x, y} = \sqrt {\sum_{i \mathop = 1}^{n + 1} \paren {x_i - y_i}^2}$

where $x = \tuple {x_1, x_2, \ldots, x_{n + 1} }, y = \tuple {y_1, y_2, \ldots, y_{n + 1} }$.

Then $\struct {\Bbb S^n, d_S}$ is a metric subspace of $\struct {\R^{n + 1}, d}$, where $d$ is the Euclidean metric on the real vector space $\R^{n + 1}$.