Fourier inversion

Question

For $f\in L^1(\mathbb R^d)$ look at $f_n:\mathbb R^d \rightarrow \mathbb K, x\mapsto\int_\mathbb {R^d}\hat{f}(\xi)e^{2\pi i\langle\xi, x\rangle} e^{-\frac{\pi^2}{n}|\xi|^2}d\xi$.
Show $\lim\limits_{n\to\infty} \|f_n-f\|_1=0$
$\hat{f}(x)$ is the Fourier transformation $\hat{f}(\xi)=\int_{\mathbb R^d}f(x)e^{-2\pi i\langle x,\xi\rangle}dx$
How can I show this? I know the Fourier transformation of $g_n(x)=e^{-\frac{\pi^2}{n}|\xi|^2}$ is $\hat{g}_n(\xi)=\frac{n^{d/2}}{\pi^{d/2}}e^{-n|\xi|^2}$ with $\|g\|_1=1$. Maybe I can use Fubine or Lebesgue dominated convergence theorem?
Thanks for a hint.