In the proof, we assume ##(0,1)## to be countable and we write \begin{align*}a_0&=0.a_{00}a_{01}a_{02}\ldots \\ a_1&=0.a_{10}a_{11}a_{12}\ldots \\ a_2&=0.a_{20}a_{21}a_{22}\ldots \\ &\vdots\end{align*}and so on for the elements ##a_0,a_1,\ldots## in ##(0,1)## (and where the expansions do not end...