Example of an irrational number of this form.

Question

Let $b:=\overline{0,b_{1},…}$ such that $b_{k} \in \{1,2\}$ and $b$ is irrational. Is $0,121122111222…$ a good example? Can you give me another example of number like this?

Answer 1

It is correct, since that decimal expansion is not almost-periodic. Another possibility would be $0.121221222122221\ldots$.