The Fourier transform of Dirac delta is often naively calculated by considering Delta function as a function that makes sense within an integral and by using its fundamental property:
$$ \int_{-\infty}^{+\infty} f(x) \delta(x-x_0)dx = f(x_0) $$
Then the Fourier transform of Dirac delta function can be evaluated to be $\hat{\delta(p)} = \frac{1}{\sqrt{2\pi}}$ simply by applying such property to the definition of Fourier transform.
However, here I am seeking for a formal proof using theory of distributions. In particolar, we know that the succession $D_n(x) := nD(nx)$ approximates Dirac delta $\delta(x)$, with $D(x) = \frac{1}{\sqrt{\pi}} e^{-x^2}$. My objective is to derive the same result above discussed using this approach.
The Fourier transform of a Gaussian-like function is known from theory, therefore it is straightforward that:
$$ \mathcal{F} [ D(x) ] = \mathcal{F} \left[ \frac{e^{-x^2}}{\sqrt{\pi}} \right] = \frac{e^{-\frac{p^2}{4}}}{\sqrt{2\pi}} $$
But this doesn't itself give me directly the solution, as there's still the term $e^{-\frac{p^2}{4}}$; how should I proceed to prove my thesis formally?