I have a set P in R constructed as follows:
Let $E_0 = [0,1]=$ {$.d_1d_2... : 0\leq d_j \leq 9 $ for all $j$}.
Let $E_1 = ${$x \in E_0 : d_1 \ne 0$}
Let $E_2 = ${$x \in E_1 : d_2 \ne 0$}
Continue in this way and define $E_0,E_1,...,E_n$ so that $E_n =${$ x \in E_{n-1}: d_n \ne 0$}
Define the set P =$ \bigcap_{j=0}^{\infty} E_j$
What is the length of P?
I don't know if I could numerate the element of P. I tried numerating then so I got something like this, which I don't think whether is correct.
$E_0 = 010101...$
$E_1 = 101010...$
$E_2 = 110101...$
Now I'm not sure about what exactly the length is.