Question 22
$x_0$ is a real number in the interval $(0, 1)$ with decimal representation $0.d_1 d_2 d_3 \ldots$. We obtain the sequence $x_n$ as follows. $x_{n+1}$ is obtained from $x_n$ by rearranging the 5 digits $d_{n+1}$, $d_{n+2}$, $d_{n+3}$, $d_{n+4}$, $d_{n+5}$. Show that the sequence $x_n$ converges. Can the limit be irrational if $x_0$ is rational? Find a number $x_0$ so that every member of the sequence is irrational, no matter how the rearrangements are carried out.