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

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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:

$\displaystyle \forall x, y \in \Bbb S^n: d_S \left({x, y}\right) = \sqrt {\sum_{i \mathop = 1}^{n + 1} \left({x_i - y_i}\right)^2}$

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


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


Proof


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