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

Then $d$ is indeed a metric.

Proof
Recall that $\set {r_j}_{j \mathop \in \N}$ is an enumeration of the rational numbers $\Q$.

Also, we note that:

meaning that this is a convergent series and so the definition is meaningful.

Next the axioms for a metric are checked in turn.

Proof of $\text M 2$
Let $i \in \N$ be fixed.

Define:
 * $\map {f_i} x := \displaystyle \max_{j \mathop \le i} \frac 1 {\size {x - r_j} }$

Then:

Now since $\inf \set {1, \size {\map {f_i} x - \map {f_i} z} } \le 1$, it follows that:

Suppose now that $\size {\map {f_i} x - \map {f_i} z} \le 1$.

Then:

On the other hand, if:
 * $\size {\map {f_i} x - \map {f_i} z} > 1$

then also:
 * $\size {\map {f_i} x - \map {f_i} y} + \size {\map {f_i} y - \map {f_i} z} > 1$

and:

Combining both cases with the estimates above, we conclude:


 * $\displaystyle \inf \set {1, \size {\map {f_i} x - \map {f_i} z} } \le \inf \set {1, \size {\map {f_i} x - \map {f_i} y} } + \inf \set {1, \size {\map {f_i} y - \map {f_i} z} }$

Finally, now, we have:

Proof of $\text M 3$
We have that:

Hence:

Proof of $\text M 4$
Suppose that $x \ne y$.

Then: