# How to prove $\mathcal H$ with VC dimension $d$ shatter all subsets with size less than $d-1$?

I was wondering that if a certain hypothesis class $$H$$ has a VC dimension $$d$$ over domain $$X$$ how to prove that $$H$$ will shatter all subsets of $$X$$ with size less than $$d$$ i.e $$H$$ 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 the definition, $$\mathcal V \mathcal C$$ dimension of $$\mathcal H$$ over domain $$X$$ is $$d=2$$. Although, $$A = \{3\} \subset X$$, whose size is smaller than the $$\mathcal V \mathcal C$$ dimenion i.e $$|A| it is not shattered by $$\mathcal H$$.