Rational Numbers in Real Euclidean Plus Space are Open Set

Theorem
Let $\R$ be the set of real numbers.

Let $d: \R \times \R \to \R$ be the Euclidean plus metric:
 * $d \left({x, y}\right) := \left\vert{x - y}\right\vert + \displaystyle \sum_{i \mathop = 1}^\infty 2^\left({-i}\right) \inf \left({1, \left\vert{ \max_{j \mathop \le i} \frac 1 {\left\vert{x - r_j}\right\vert} - \max_{j \mathop \le i} \frac 1 {\left\vert{y - r_j}\right\vert} }\right\vert }\right)$

Let $\Q$ be the set of rational numbers.

Then $\Q$ is an open set of $\left({\R, d}\right)$.