Let $T$ an distribution defined by $$\varphi \in \mathcal{D}(\mathbb{R}):=\sum_{m=1}^{+\infty}(\varphi(\dfrac{1}{m})-\varphi(0)-\dfrac{\varphi'(0)}{m})$$ We have that support of $T$ is $\operatorname{Supp}(T)= \{0\} \cup \Bigl\{\dfrac{1}{m}: m\in \mathbb{N}^*\Bigr\}$, then we consider the sequence $\varphi_m$ such as $\varphi_m(x)=\dfrac{1}{\sqrt{m}}$ neighborhood $\Bigl[\dfrac{1}{m},1\Bigr]$ and $\varphi_m(x)= 0$ neighborhood $\Bigl[0,\dfrac{1}{m+1}\Bigr]$ and such as $0 \leq \varphi_m \leq 1$. We have that $\lim_{m \to +\infty} T(\varphi_m) =+\infty$.
The question is: how to prove that $T$ don't satisfy the equality $$ T(\varphi)= C \sup_{|\alpha|\leq k} \sup_{\operatorname{Supp}(T)} |D^\alpha \varphi| $$ I try to calculate $\sup_{|\alpha|\leq k} \sup_{\operatorname{Supp}(T)} |D^\alpha \varphi_m|$, but I have difficulties.