Space generated by Gaussian functions

Question

Let A be the space generated by all Gaussians functions $$f_{m, \sigma}(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-m)^2}{2\sigma^2}}$$ where $m , \sigma \in \mathbb{R}$. Let B be the space of all continuous function of $x$ on $\mathbb{R}$ that converges to zero as $x$ tends to infinity. I'm wondering if A is dense in B by norm $\|\|_{\infty}$.