another example of Dirac sequence

Let $A_n=[{\displaystyle \frac{-1}{2^n}},{\displaystyle \frac{1}{2^n}}]$ and ${\delta }_n=2^{n-1}{\chi }_{A_n}$ for every positive integer $n$, where ${\chi }_S$ denotes the characteristic function of the set $S$. Then $\{{\delta }_n\}$ is a Dirac sequence:

1. ${\delta }_n(t)\ge 0$ for every positive integer $n$ and every $t\in ℝ$.

2. Let $n$ be a positive integer. Then ${\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}}{\delta }_n(t)𝑑t={\displaystyle \underset{\frac{-1}{2^n}}{\overset{\frac{1}{2^n}}{\int }}}{2}^{n-1}𝑑t=1$.

3. Let $r>0$. Then there exists a positive integer $N$ such that, for every integer $k>N$, we have . Thus, for every integer $k>N$, we have ${\displaystyle \underset{ℝ\setminus [-r,r]}{\int }}{d}_k(t)𝑑t=0$.