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I have a set P in R constructed as follows:

  1. Let $E_0 = [0,1]=$ {$.d_1d_2... : 0\leq d_j \leq 9 $ for all $j$}.

  2. Let $E_1 = ${$x \in E_0 : d_1 \ne 0$}

  3. Let $E_2 = ${$x \in E_1 : d_2 \ne 0$}

  4. 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$}

  5. 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.

2 Answers 2