Let $(a_n)_n\subseteq \Bbb{R}$ such that $$\left\langle T,\varphi \right\rangle=\sum^{\infty}_{n=0}a_n\varphi(n).$$ I want to prove that $T\in D'(\Bbb{R}).$
My trial
It suffices to prove that $T\in l^1_{loc}(\Bbb{R})$. That is, $$ \sum^{\infty}_{n=0}\left|a_n\varphi(n)\right|<\infty.$$ Let $\varphi\in D(\Bbb{R})$, then there exists $a>0$ such that $\text{supp}\varphi\subseteq [-a,a]$, where supp is the support of $\varphi$.
$$\left|\left\langle T,\varphi \right\rangle\right|\leq \sum^{\infty}_{n=0}\left|a_n\right| \left|\varphi(n)\right|\leq \sup_{n\in [-a,a]} \left|\varphi(n)\right|\sum^{\infty}_{n=0}\left|a_n\right|.$$ Since $(a_n)_n\subseteq \Bbb{R}$, I am not sure that $\sum^{\infty}_{n=0}\left|a_n\right|<\infty.$ So, I am stuck here, as I don't know how to proceed. Any help, please?