Let $\{x_{j}\}_{j\in \mathbb{N}}\subset\Omega$ a set without accumulation points in $\Omega$. For each $j\in \mathbb{N}$ consider the multi index $\alpha_{j}\in \mathbb{Z}_{+}^{n}$ define $$u(\varphi)=\sum_{j\in\mathbb{N}}\partial^{\alpha_{j}}\varphi(x_{j}), \forall\varphi\in C_{0}^{\infty}(\Omega).$$
$(a)$ Show that $u\in\mathcal{D}^{\prime}(\Omega).$
$(b)$ Show that $u\in\mathcal{D}^{\prime}_{F}$ if, and only if, exist $M>0$ such that $|\alpha_{j}|\leq M,$ for all $j\in\mathbb{N}.$
$(c)$ If $u\in \mathcal{D}^{\prime}_{F}(\Omega),$ then $u\in\mathcal{D}^{\prime}_{m}(\Omega),$ where $m=\max|\alpha_{j}|.$
I did the letter (a), could someone help me with the letters $(b)$ and $(c),$ please?