Proof that $(\Bbb{Z},d/n)$ tends to $\Bbb{R}$ in the Gromov-Hausdorff sense

Question

Where can I find the full proof that $(\Bbb{Z},d/n)$ tends for $n\to\infty$ to $(\Bbb{R},d)$ in the Gromov-Hausdorff sense, where $d(x,y):=|x-y|$? Or at least, where is it proven that such a sequence is Cauchy?

Answer 1

You can find it right here... the function $f_n(k)=k/n$ is an isometric embedding of $(\mathbb{Z},d/n)$ into $\mathbb{R}$. Its image is the set $A = \{k/n:k\in \mathbb{Z}\}$. The complement of this set consists of open intervals of length $1/n$. Hence, every point of $\mathbb{R}$ is within distance at most $1/(2n)$ from the set $A$. Since every point of $A$ is also a point of $\mathbb{R}$, we have $d_H(A,\mathbb{R}) \le 1/(2n)$ by the definition of Hausdorff distance.

Recalling the definition of Gromov-Hausdorff metric (with infimum over all isometric embeddings, etc) we see that $$d_{GH}((\mathbb{Z}, d/n), \mathbb{R}) \le d_H(A,\mathbb{R}) \le \frac{1}{2n}\to 0$$ as claimed.