Euclidean Plus Metric is Metric

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

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

Then $d$ is indeed a metric.