# How do I prove that $\mathcal{H}$, with $\mathcal{VC}$ dimension $d$, shatters all subsets with size less than $d-1$?

If a certain hypothesis class $$\mathcal{H}$$ has a $$\mathcal{VC}$$ dimension $$d$$ over a domain $$X$$, how can I prove that $$H$$ will shatter all subsets of $$X$$ with size less than $$d$$, i.e. $$\mathcal{H}$$ will shatter $$A \subset X$$ where $$|A| \leq d-1$$?

We can show that it is not true by a counterexample. For example, $$X = \{1,2,3\}$$ and $$\mathcal H = \{\{\},\{1\},\{2\},\{1,2\}\}$$ is the finite set hypothesis class. By definition, in this case, the $$\mathcal{VC}$$ dimension of $$\mathcal H$$ over the domain $$X$$ is $$d=2$$. Although $$A = \{3\} \subset X$$, whose size is smaller than the $$\mathcal{VC}$$ dimension, i.e $$|A|, it is not shattered by $$\mathcal H$$.