![a_{0},\ a_{1},\ a_{2},\dots](/media/m/2/4/3/243fe45a15301566848c13d2144aa585.png)
is a sequence of real numbers such that
prove that exist
![j](/media/m/7/9/e/79ebb10f98eb80d16b0c785d9d682a72.png)
such that for every
![i\geq j](/media/m/e/a/d/ead3eea9e310782c9b34fb761e4ca953.png)
we have
![a_{i + 2} = a_{i}](/media/m/8/7/b/87bc8009fe90111f7cabc75c8e275a5f.png)
.
%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 $a_{i + 2} = a_{i}$.