One of my daughters was having a small programming exercise.

Let's consider following algorithm:

- Take a list of length $n$: $\ (1\,\ 2\,\ \ldots\,\ n)$.
- Remove every $2$nd number.
- From the resulting list, remove every $3$rd number.
- From the resulting list, remove every $4$th number.
- ... Follow on until the list remains unchanged and let $u_n$ be the number of remaining elements.

Example with $n=11$

- $(\ 1\,\ 2\,\ 3\,\ 4\,\ 5\,\ 6\,\ 7\,\ 8\,\ 9\,\ 10\,\ 11\ )\quad \Rightarrow\quad (\ 1\ *\ 3\ *\ 5\ *\ 7\ *\ 9 \ *\ 11\ )$
- $(\ 1\,\ 3\,\ 5\,\ 7\,\ 9\,\ 11\ )\quad \Rightarrow\quad (\ 1\,\ 3\ *\ 7\,\ 9\ *\ )$
- $(\ 1\, 3\,\ 7\,\ 9\ )\quad \Rightarrow\quad (\ 1\,\ 3\,\ 7\ *\ )$
- $(\ 1\,\ 3\,\ 7\ )\ $ -- will not be modified anymore, and therefore $u_n=3$.

**QUESTION:** why do we have $\lim\limits_{n \to +\infty} \frac{n}{u_n^2}=\frac{\pi}{4}$ ?

Thanks!