I have the following exercise.
Let $T: \mathcal{D}(\mathbb{R}) \to \mathbb{C}$ defined by $$ T: \varphi \to \langle T,\varphi \rangle = \sum_{m=1}^{+\infty} (\varphi(\dfrac{1}{m})-\varphi(0)- \dfrac{\varphi'(0)}{m}) $$ 1. Prove that $T$ is an distribution.
Prove that $Supp(T)=\{0\} \cup \{\dfrac{1}{m}: m \in \mathbb{N}^{\star}\}$.
For $m \in \mathbb{N}^\star$ we consider a function $\varphi_m$ such as $\varphi_m=\dfrac{1}{\sqrt{m}}$ neighborhood $[\dfrac{1}{m},1]$ and $0$ neighborhood $[0,\dfrac{1}{m+1}]$ such as $0 \leq \varphi_m \leq \dfrac{1}{\sqrt{m}}$. Prove that there exists $\varphi$ such as $\varphi_m \to \varphi$ in $\mathcal{D}(\mathbb{R})$.
Prove that $\lim_{n \to +\infty} T(\varphi_m)=+\infty$.
Deduce that there exists an distribution $T$ with compact support such as it now satisfies the relation $$T(\varphi)= C \sup_{|\alpha|\leq k} \sup_{Supp(T)} |D^\alpha (\varphi)|$$
I have difficulties to reslves questions $3$ and $5$. Can you help me please mostly in question $3$.