Is dirac comb a tempered distribution? Prove or disprove the statement
$$ III:=\sum_{k=-\infty}^\infty\delta_k\in \mathscr{S}'(\mathbb{R}^n)$$
where $\delta_k\varphi:=\varphi(k),\,\,\forall\varphi\in\mathscr{S}(\mathbb{R}^n)$
I tried to prove linearity and I think it holds, but I am not so sure about the continouity.

I think yes; if $p_n(x)$ converges to $0$ in the Schwartz space, then in particular, for any $\alpha>0$ the sequence of numbers $\sup_x |x^\alpha p_n(x)|$ converges to $0$ for $n\to\infty$. 
In particular, for any $\epsilon>0$ there exists $N$ s.t. for $n>N$, $|p_n(x)|$ is smaller then $\epsilon$ on $x\in [-1,1]$ and smaller than $\epsilon x^{-2}$ for $x\in \mathbb{R}\setminus [-1,1]$. But then $\sum_{k\in\mathbb{Z}} p_n(k)$ is smaller than $\epsilon (3 + \frac{\pi^2}{3})$ or something like that, so $III(p_n)$ converges to $0$.
