Školjka

%V0 $a_{0},\ a_{1},\ a_{2},\dots$ is a sequence of real numbers such that $$a_{n + 1} = \left[a_{n}\right]\cdot \left\{a_{n}\right\}$$ prove that exist $j$ such that for every $i\geq j$ we have $a_{i + 2} = a_{i}$.