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Non-empty, compact, disconnected and limit points

I am at the moment trying to get through some basic set theory and I'm getting very stuck with the proofs. This question is from a textbook I am studying from and as it is a prove question there is no solution in the back :(

Any help with explanations would be very useful

Let Attachment 23327 where *E*0 = [0*, *1] and *Ek *is constructed by removing the second, fourth and sixth sevenths of each component Ek-1. Each set that is removed is open.

I need to show that F is non-empy, compact, totally disconnected and that 0 is a limit point of F

Re: Non-empty, compact, disconnected and limit points

Show that each $\displaystyle E_k$ is compact, which will give you the first two points. For totally disconnectedness, show that $\displaystyle E$ doesn't contain any interval. Show that for each $\displaystyle k$ there is $\displaystyle x_k$ such that $\displaystyle |x_k|\leq 2^{-1}\cdot 7^{-k}$.