Question 1
A sequence of real numbers $a_{0}, a_{1}, a_{2}, \ldots$ is defined by the formula
$$
a_{i+1}=\left\lfloor a_{i}\right\rfloor \cdot\left\langle a_{i}\right\rangle \quad \text{ for } \quad i \geq 0
$$
here $a_{0}$ is an arbitrary real number, $\left\lfloor a_{i}\right\rfloor$ denotes the greatest integer not exceeding $a_{i}$, and $\left\langle a_{i}\right\rangle=a_{i}-\left\lfloor a_{i}\right\rfloor$. Prove that $a_{i}=a_{i+2}$ for $i$ sufficiently large.
$$
a_{i+1}=\left\lfloor a_{i}\right\rfloor \cdot\left\langle a_{i}\right\rangle \quad \text{ for } \quad i \geq 0
$$
here $a_{0}$ is an arbitrary real number, $\left\lfloor a_{i}\right\rfloor$ denotes the greatest integer not exceeding $a_{i}$, and $\left\langle a_{i}\right\rangle=a_{i}-\left\lfloor a_{i}\right\rfloor$. Prove that $a_{i}=a_{i+2}$ for $i$ sufficiently large.