Open Ball in Euclidean Plus Metric is Subset of Equivalent Ball in Euclidean Metric

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 $d': \R \times \R \to \R$ be the Euclidean metric.

Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.

Let $\map {B_\epsilon} {p; d}$ be an open $\epsilon$-ball of $p$ in $\R$ on $d$.

Let $\map {B'_\epsilon} {p; d'}$ be an open $\epsilon$-ball of $p$ in $\R$ on $d'$.

Then:
 * $\map {B_\epsilon} {p; d} \subseteq \map {B'_\epsilon} {p; d'}$

Proof
Let $p \in \R$.

Let $\epsilon \in \R_{>0}$.

Let $x \in \map {B_\epsilon} {p; d}$.

Then:

Thus:
 * $x \in \map {B_\epsilon} {p; d} \implies x \in \map {B_\epsilon} {p; d'}$

and the result follows by.