Please help me prove this question:
For the range space $(\Bbb R^d, D_d)$ where $D_d$ consists all the open disks in $\Bbb R^d$, each disk in $D_d$ is in the form of: $D(p,r):=\{x\in\Bbb R^d | \Vert x-p\Vert_2<r\}$, for $p\in\Bbb R^d $ and $r>0$.
Show that $\operatorname{VCdim}(\Bbb R^d, D_d) \le d+2$.
There are two ways to solve this (that I know of), one of them is explained here: https://cstheory.stackexchange.com/a/11089. This proof is for for closed disks not open, but we can change this proof to consider two different cases, one of them is that there are points on the edge of the ball B (i.e. points such that $||x-p||_2=r$) and the other that there aren't such points, if there aren't any points on the edge their proof doesn't change, if there are we shrink $r$ a little bit such that the points on the edge are no longer in the ball but we also didn't remove any other points from B, and now we continue the proof without changes.
And a different way to proof this is this: Let S be the largest set of points such that $2^{|S|}=|\Pi(S)|$. For some fixed p,r let $T=\{x|x\in S,||x-p||_2<r\}$ and let $S'$ be the minimum enclosing hypersphere (d-dimentional ball) of the points of $T$. A point $x\in S$ satisfies $||x-p||_2<r$ if and only if $x\in S'$. The minimum enclosing sphere of a set of points can be determined by at most d+1 points from the set i.e. there is a subset of size at most d+1 such that it's minimum enclosing sphere is the minimum enclosing sphere of all the points (see Lemma 3.7 from https://www.google.com/url?sa=t&source=web&rct=j&url=https://people.inf.ethz.ch/emo/DoctThesisFiles/fischer05.pdf&ved=2ahUKEwjs8ZnO16DuAhUWH-wKHVqSBKUQFjAAegQIARAB&usg=AOvVaw0lnz91H2v8Oqa1oPvgyk6Y) . From that we get that there exists a subset of at most d+1 points from $T$ (and therefore also from S) that defines for all the rest of the points whether or not their distance from p is at most r. Meaning that:
$$2^{|S|}=|\Pi(S)|\leq \sum_{i=0}^{d+1}{\binom{|S|}{i}}$$ Now from the binomial theorem we get: $$\sum_{i=0}^{|S|}{\binom{|S|}{i}}\leq \sum_{i=0}^{d+1}{\binom{|S|}{i}}$$
Meaning that $VCdim(R^d,D_d)=|S|\leq d+1$
Sorry for bad english.