Vector Subspace of Real Vector Space under Chebyshev Metric is Metric Subspace

Theorem

Let $n \in \N$.

Let $A$ be the set of all ordered $n+1$-tuples $\left({x_1, x_2, \ldots, x_{n+1} }\right)$ of real numbers such that $x_{n+1} = 0$.

Let $d: A \times A \to \R$ be the function defined as:

$\displaystyle \forall x, y \in A: d \left({x, y}\right) = \max_{i \mathop = 1}^n \left\{ {\left\lvert{x_i - y_i}\right\rvert}\right\}$

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

Then $\left({A, d}\right)$ is a metric subspace of $\left({\R^{n+1}, d_\infty}\right)$ where $d_\infty$ is the Chebyshev distance on the real vector space $\R^{n+1}$.