For sake a clarity, when I say a "limit point", the definition I am using is:
$x$ is a limit point of a set $A$ if there exists a sequence $\{x_k\}$ such that $x_k \neq x$ and $x_k \rightarrow x$ as $k \rightarrow \infty$
(as opposed to the more general definition about a topological space).
My set up is as follows: Let $E_0 = [0,1]$ and $E_{k+1}$ be constructed by removing the second, fifth and eighth tenths of $E_k$. Then define $F = \cap_{k=0}^{\infty} {E_k}$.
I am trying to show that $0$ (zero) is a limit point of $F$.
As with these types of questions, I can see the intuition behind why this is true, but I'm struggling to actually prove it is true. In particular, I'm struggling to figure out how to construct the sequence $\{x_k\}$.
My first thought was to try and construct the sequence using the maximum point of each $E_k$ at each step, but realised this wouldn't work for a variety of reasons; mainly that the points in this sequence were not in $F$ itself. I guess what I'm struggling with most is how to formally construct the sequence, since I can't think of a `nice' way to define it properly.
Any help about how to construct this sequence, or a different method to try, would be greatly appreciated!