Example of tightness of Sauer-Shelah lemma

I was given the task of finding an example of a family of events (or hypothesis, concepts, etc.) such that the Sauer-Shelah lemma is tight.

The lemma states:

Assume that the Vapnik-Chervonenkis dimension (VC-dim) of a family of events $$\mathcal{A}$$ is $$d$$. Then for $$n \geq d$$ $$\mathcal{S}_\mathcal{A}(n) \leq \sum_{i=0}^d{n \choose i}$$ Where $$\mathcal{S}_\mathcal{A}$$ is the shatter function.

My solution: let $$\mathcal{X} = \mathbb{R}$$ and $$\mathcal{A} = \{x\}_{x \in \mathcal{X}}$$. My understanding of VC-dim tells me that $$\mathcal{A}$$ has VC-dim = 1, since we are not able to shatter more than one point with sets in $$\mathcal{A}$$.

Now, choose a set with $$n$$ points $$S =\{x_1, ..., x_n\}$$. Notice there are $$n+1$$ different ways to shatter this set using $$\mathcal{A}$$ (choose each point once and then choose a point $$x \not\in S$$). Moreover,

$$n+1 = \sum_{i=0}^1{n \choose i} = \underbrace{n \choose 0}_1 + \underbrace{{n \choose 1}}_n$$

Is this example correct ?

Your justification for the VC-dim of $$\mathcal{A}$$ being $$1$$ is correct and the example is fine.
As a note, you can generalize your example by considering $$\mathcal{A}$$ to be the subsets of $$\mathbb{R}$$ with $$d$$ elements. This family has VC-dim $$d$$ and given a subset $$S = \{x_1, x_2, ..., x_n\}$$, where $$n > d$$, there are exactly $$\sum_{i=0}^d {n\choose i}$$ ways in which $$d$$ points can shatter $$S$$. Thus,
$$\mathcal{S}_\mathcal{A}(n) = \sum_{i=0}^d {n\choose i}.$$