Question 54
A sequence $\left(u_{n}\right)_{n \geqslant 0}$ of positive integers is called Sicilian if
$$
u_{n+1} \in\left\{u_{n} / 2, u_{n} / 3, 2 u_{n}+1, 3 u_{n}+1\right\}$$
for every $n \geqslant 0$. Prove that for every integer $k \geqslant 1$, there exist a Sicilian sequence $\left(u_{n}\right)_{n \geqslant 0}$ and an integer $\ell \geqslant 0$ such that $u_{0}=k$ and $u_{\ell}=1$.
$$
u_{n+1} \in\left\{u_{n} / 2, u_{n} / 3, 2 u_{n}+1, 3 u_{n}+1\right\}$$
for every $n \geqslant 0$. Prove that for every integer $k \geqslant 1$, there exist a Sicilian sequence $\left(u_{n}\right)_{n \geqslant 0}$ and an integer $\ell \geqslant 0$ such that $u_{0}=k$ and $u_{\ell}=1$.