Gaussian Delta Sequence

Theorem
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:


 * $\ds \map {\delta_n} x := \frac n {\sqrt \pi} e^{- n^2 x^2}$

Then $\sequence {\map {\delta_n} x}_{n \mathop \in {\N_{>0} } }$ is a delta sequence.

That is, in the distributional sense it holds that:


 * $\ds \lim_{n \mathop \to \infty} \map {\delta_n} x = \map \delta x$

or


 * $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x = \map \delta \phi$

where $\phi \in \map \DD \R$ is a test function, $\delta$ is the Dirac delta distribution, and $\map \delta x$ is the abuse of notation, usually interpreted as an infinitely thin and tall spike with its area equal to $1$.