Theorem: Tuning Fork Delta Sequence

thumb300pxThe graph of the tuning fork delta sequence. As $n$ grows, the rectangles becomes thinner and longer. The area of each shape is equal to $1$, where the area under the axis contributes negatively. Note that at $x = 0$ the graph is always negative, and its value here approaches $-\infty$. Only taking the entire graph we can ensure the proper behaviour for $n$ approaching $\infty$,
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:
:<nowiki>$\map {\delta_n} x := \begin {cases}
-n & : \size x < \dfrac 1 {2n} \\ \\
2n & : \dfrac 1 {2n} \le \size x \le \dfrac 1 n \\ \\
0 & : \size x > \dfrac 1 n \end {cases}$</nowiki>
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
:$\map \delta x$ 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:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x
      | r = \lim_{n \mathop \to \infty} \paren {\int_{- \infty}^{-\frac 1 n } \map \phi x \map {\delta_n} x \rd x + \int_{- \frac 1 n }^{- \frac 1 {2n} } \map \phi x \map {\delta_n} x \rd x + \int_{- \frac 1 {2n} }^{\frac 1 {2n} } \map \phi x \map {\delta_n} x \rd x + \int_{\frac 1 {2n} }^{\frac 1 n} \map \phi x \map {\delta_n} x \rd x + \int_{\frac 1 n}^{\infty} \map \phi x \map {\delta_n} x \rd x}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \paren {2n \int_{- \frac 1 n }^{- \frac 1 {2n} } \map \phi x \rd x - n \int_{- \frac 1 {2n} }^{\frac 1 {2n} } \map \phi x \rd x + 2n \int_{\frac 1 {2n} }^{\frac 1 n} \map \phi x \rd x}
}}
{{eqn | r = \lim_{n \mathop \to \infty} n \paren {2 \map \phi {\xi_1} \paren {- \frac 1 {2n} - \paren {- \frac 1 n} } - \map \phi {\xi_2} \paren {\frac 1 {2n} - \paren {- \frac 1 {2n} } } + 2 \map \phi {\xi_3} \paren {\frac 1 n - \frac 1 {2n} } }
      | c = Mean Value Theorem for Integrals, $\xi_1 \in \closedint {-\frac 1 n } {-\frac 1 {2n} }$, $\xi_2 \in \closedint {-\frac 1 {2n} } {\frac 1 {2n} }$, $\xi_3 \in \closedint {\frac 1 {2n} } {\frac 1 n }$
}}
{{eqn | r = \lim_{n \mathop \to \infty} \paren { \map \phi {\xi_1} - \map \phi {\xi_2} + \map \phi {\xi_3} }
}}
{{eqn | r = \map \phi {\lim_{n \mathop \to \infty} \xi_1} - \map \phi {\lim_{n \mathop \to \infty} \xi_2} + \map \phi {\lim_{n \mathop \to \infty} \xi_3}
      | c = Limit of Image of Sequence on Real Numbers
}}
{{eqn | r = \map \phi 0 - \map \phi 0 + \map \phi 0
      | c = $\ds \lim_{n \mathop \to \infty} \frac 1 n = 0$, Squeeze Theorem for Real Sequences
}}
{{eqn | r = \map \delta \phi
      | c = {{Defof|Dirac Delta Distribution}}
}}
{{end-eqn}}
Furthermore:
{{begin-eqn}}
{{eqn | l = \int_{-\infty}^\infty \map {\delta_n} x \rd x
      | r = 2n \int_{- \frac 1 n }^{- \frac 1 {2n} } \rd x - n \int_{- \frac 1 {2n} }^{\frac 1 {2n} } \rd x + 2n \int_{\frac 1 {2n} }^{\frac 1 n} \rd x
}}
{{eqn | r =  n \paren {2 \paren {- \frac 1 {2n} - \paren {- \frac 1 n} } - \paren {\frac 1 {2n} - \paren {- \frac 1 {2n} } } + 2 \paren {\frac 1 n - \frac 1 {2n} } }
}}
{{eqn | r =  n \paren {\frac 2 {2n} - \frac 2 {2n} + \frac 2 {2n} }
}}
{{eqn | r =  1
}}
{{end-eqn}}
{{qed}}