Convergence in $\mathcal{D}'(\mathbb{R})$ I want to make a statement about the following sequence of functions whether it converges in $\mathcal{D}'(\mathbb{R})$ (space of distributions):
\begin{align}
\phi_n(x) = e^{-x^2/n^2}.
\end{align}
I want to do it as analytical as possible.
The definition is clear but I do not see how to applicate it to our problem:
$T_n \rightarrow T$ in $\mathcal{D}'(\mathbb{R})$ if $T_n(f) \rightarrow T(f)$ for all $f \in \mathcal{D}(\mathbb{R})$.
Is there anyone who can help me :)

For any $f \in \mathcal{D}(\mathbb{R})$
$$
\langle\phi_n,f\rangle=\int_{\Bbb R}e^{-x^2/n^2}\,f(x)\,dx.
$$
Use the dominated convergence theorem to show that the integral converges as $n\to\infty$. 
