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:
 * $\map d {x, y} := \size {x - y} + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$

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

Then $\Q$ is an open set of $\struct {\R, d}$.