2006 IMO Shortlist N2

For $x \in(0,1)$ let $y \in(0,1)$ be the number whose $n$th digit after the decimal point is the $\left(2^{n}\right)$ th digit after the decimal point of $x$. Show that if $x$ is rational then so is $y$. (Canada)

对于$x \in(0,1)$，设$y \in(0,1)$为小数点后第$n$位数字是$x$小数点后第$\left(2^{n}\right)$位的数字。证明如果 $x$ 是有理数，那么 $y$ 也是有理数。 (加拿大)