Consider
I understand the example without the existance part. In particular, how do we know that such $A_n$ exist? I was trying to play around with the $(0,1)$ interval but this looks fruitless. Next I thought that I can just consider all over $\mathbb R$ with the Lebesgue measure with some density, say $\frac{e^{-x^{2}}}{\sqrt{\pi }}$. Then let \begin{align*} A_n = [n, n + \gamma_n ] \end{align*} where $\gamma _n$ is such that \begin{align*} \frac{1}{n} = \int_{n}^{n + \gamma_n} \frac{e^{-x^{2}}}{\sqrt{\pi }} dx \end{align*} I think this works but I am not sure. This seems far too involved. Am I missing something?
