MathNet 2007-part-01 — 187 problems (EN–VI) Question 175

For every real number $x$ with $0 \lt x \lt 1$ let $y \in (0, 1)$ be the number whose $n$-th decimal digit (after the decimal point) is the $2^{n}$-th decimal digit of $x$. Prove: if $x$ is rational, then $y$ is also rational.