A sequence of functions converging to the Dirac delta

Question

let $g_n(x)=\frac{1}{2}n $ for $|x|<\frac{1}{n}$ and for positive integer n. Prove that $$\lim_{n \to \infty} g_n(x)=\delta(x)$$

Pretty evident after a quick sketch, but I don't know how to show it.

Answer 1

There is a typo in the question. Should be $n \to \infty$

Let $f(x)$ be any function that is continuous in the neighborhood of $0$. Suppose that $n$ is large enough so that $|x| < 1/n$ is inside this interval. Then $$ \int_{-\infty} ^{\infty} g_n(x) f(x) = \int_{-\frac1n} ^{\frac1n} g_n(x) f(x) =\frac 1 2 n \int_{-\frac1n} ^{\frac1n} f(x)=\frac 1 2 n \cdot \left( \frac 2 n f(\theta)\right) $$ for some $\theta$ in the interval $[-1/n, 1/n]$. As $n\to \infty$, $\theta$ gets squeezed to zero and hence $$ \lim_{n\to \infty} \int_{-\infty} ^{\infty} g_n(x) f(x) = f(0)$$ which is the definition of the delta function.