TITLE: Is it always possible to "encircle" exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?
QUESTION [9 upvotes]: Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there always a point $p \in \mathbb{R}^d$ and a radius
$r \in \mathbb{R}^+$ such that the $r$-neighbourhood of $p$ contains exactly $n$
points in $S$?
If the answer is yes, what can be said in general about in which metric spaces this holds?

REPLY [13 votes]: Yes, this is possible.
Pick $p$ generic, so that
$$\not \exists\,\,\, x,y\in S\quad\text{s.t.}\quad d(p,x)=d(p,y).$$
Then slowly increase the radius until the $r$-neighborhood of $p$ contains exactly $n$ points of $S$.