$$$$

Define $$f_N(x)={\displaystyle \frac{N}{\pi }}{\displaystyle \frac{1}{1+(Nx{)}^2}}$$ for all $N\in \mathbb{N}$ and all $x\in \mathbb{R}$.

1. Calculate the indefinite Riemann integral $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{f}_N(x)\mathsf{d}x=\underset{T\to \mathrm{\infty }}{lim}\underset{-T}{\overset{T}{\int }}{f}_N(x)\mathsf{d}x$$ for each $N\in \mathbb{N}$.
2. Prove that $\underset{N\to \mathrm{\infty }}{lim}{f}_N(x)=0$ for all $x\ne 0$.

Fix a continuous function $g:\mathbb{R}\to \mathbb{R}$ that is bounded.

3. Prove that $$\underset{N\to \mathrm{\infty }}{lim}\underset{\delta }{\overset{\mathrm{\infty }}{\int }}{f}_N(x)g(x)\mathsf{d}x=0$$ for every $\delta >0$.

4. Prove that $$\underset{N\to \mathrm{\infty }}{lim}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{f}_N(x)g(x)\mathsf{d}x=g(0)$$ by making use of 3. and the continuity of g at 0.

Define $$F_N(x)={\displaystyle \frac{1}{N}}\sum _{K=0}^{N-1}\sum _{k=-K}^K{e}^{2\pi ikx}$$ for all $N\in \mathbb{N}$ and all $x\in [0,1)$.

5. Prove for every continuous function $g:[0,1)\to \mathbb{R}$ that $$\underset{N\to \mathrm{\infty }}{lim}{\int }_0^1{F}_N(x)g(x)\mathsf{d}x=g(0)$$ by mimicking the above arguments.