Rectangular Delta Sequence

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


 * $\map {\delta_n} x := \begin{cases}

0 & : x < - \frac 1 {2n} \\ n & : - \frac 1 {2n} \le x \le \frac 1 {2n} \\ 0 & : x > \frac 1 {2n} \end{cases}$

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 $1$.

Proof
Let $\phi \in \map \DD \R$ be a test function.

Then: