RU-ALLRUSSIAN 1980 Problem 20

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