\[ \newcommand{\C}{\mathbb{C}} \newcommand{\haar}{\mathsf{m}} \newcommand{\P}{\mathcal{P}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\d}{\mathsf{d}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \newcommand{\B}{\mathscr{B}} \newcommand{\fou}{\mathcal{F}} \]

Define \[ f_N(x) = \dfrac{N}{\pi} \dfrac{1}{1 + (Nx)^2} \] for all $N \in \N$ and all $x \in \R$.

1. Calculate the indefinite Riemann integral \[ \int\limits_{-\infty}^\infty f_N(x) \intd x = \lim_{T \to \infty} \int\limits_{-T}^T f_N(x) \intd x \] for each $N \in \N$.
2. Prove that $\lim\limits_{N \to \infty} f_N(x) = 0$ for all $x \ne 0$.

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

3. Prove that \[ \lim_{N \to \infty} \int\limits_\delta^\infty f_N(x) g(x) \intd x = 0 \] for every $\delta \g 0$.

4. Prove that \[ \lim_{N \to \infty} \int\limits_{-\infty}^\infty f_N(x) g(x) \intd x = g(0) \] by making use of 3. and the continuity of g at 0.

Define \[ F_N(x) = \dfrac{1}{N}\sum_{K=0}^{N-1} \sum_{k=-K}^K e^{2 \pi i k x} \] for all $N \in \N$ and all $x \in [0,1)$.

5. Prove for every continuous function $g : [0,1) \to \R$ that \[ \lim_{N \to \infty} \int_0^1 F_N(x) g(x) \intd x = g(0) \] by mimicking the above arguments.