Characterization of N-Cube

Theorem
Let $\left({\R^n, d}\right)$ be a Euclidean $n$-Space equipped with the usual metric $d$.

Let $x, y \in \R^n$, where $x = \left({x_1, x_2, \ldots, x_n}\right)$ and $y = \left({y_1, y_2, \ldots, y_n}\right)$.

Fix $R > 0$.

Let:


 * $\displaystyle Q = \left\{ {x, y \in \R^n: \sup_{x, y} \max_i \left\vert{y_i - x_i}\right\vert \le R} \right\}$

Then $Q$ is an $n$-cube.

Proof
For ease of presentation, denote:


 * $y - x = r \in \R^n$

and:


 * $y_j - x_j = r_j$

for $j = 1,2, \ldots\, n$.