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:
 * $d \left({x, y}\right) := \left\vert{x - y}\right\vert + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \inf \left({1, \left\vert{ \max_{j \mathop \le i} \frac 1 {\left\vert{x - r_j}\right\vert} - \max_{j \mathop \le i} \frac 1 {\left\vert{y - r_j}\right\vert} }\right\vert }\right)$

Then $d$ is indeed a metric.

Proof
Recall that $\left\{ {r_j}\right\}_{j \mathop \in \N}$ 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, let us check the axioms for a metric in turn.

Proof of $M2$
Let $i \in \N$ be fixed.

Define:
 * $f_i \left({x}\right) := \displaystyle \max_{j \mathop \le i} \frac 1 {\left\vert{x - r_j}\right\vert}$

Then:

Now since $\inf \left\{ {1, \left\vert{f_i \left({x}\right) - f_i \left({z}\right)}\right\vert}\right\} \le 1$, it follows that:

Suppose now that $\left\vert{f_i \left({x}\right) - f_i \left({z}\right)}\right\vert \le 1$.

Then:

On the other hand, if:
 * $\left\vert{f_i \left({x}\right) - f_i \left({z}\right)}\right\vert > 1$

then also:
 * $\left\vert{f_i \left({x}\right) - f_i \left({y}\right)}\right\vert + \left\vert{f_i \left({y}\right) - f_i \left({z}\right)}\right\vert > 1$

and:

Combining both cases with the estimates above, we conclude:


 * $\displaystyle \inf \left\{ {1, \left\vert{f_i \left({x}\right) - f_i \left({z}\right)}\right\vert}\right\} \le \inf \left\{ {1, \left\vert{f_i \left({x}\right) - f_i \left({y}\right)}\right\vert}\right\} + \inf \left\{ {1, \left\vert{f_i \left({y}\right) - f_i \left({z}\right)}\right\vert}\right\}$

Finally, now, we have:

Proof of $M3$
We have that:

Hence:

Proof of $M4$
Suppose that $x \ne y$.

Then: