The book says,
No segment of the form $\left(\dfrac{3k+1}{3^{m}},\dfrac{3k+2}{3^{m}}\right)$ where $k,m\in\mathbb{Z}^{+}$ has a point in common with the Cantor set.
I can prove by mathematical induction that by the construction of Cantor set, any middle third segment has the form and has no element common with the set. However, my proof does not prove given any segment with arbitrary k and m, given that it is in [0, 1], it must be one of those middle thirds. How to prove that? Is the proof needed to prove there is no segment as such in the Cantor set?
I got the answer after checking possible positions of a segment with such form. Here is how I prove. Draw the intervals of any En, and then consider all possible positions of a segment, with the length of $\dfrac{1}{3^{m}}$. There only three cases. For the first case, it overlaps with an interval of En, excluding the end points. The second case is that it is one of the segments that are removed when the Cantor set is constructed. The last case is that part of it belongs to an interval, and part of it belongs to its adjacent interval (or segment) of P. The contradiction in the last case is obvious by checking the length of the segment, if we assume it has such form, i.e. the length is large than $\dfrac{1}{3^{m}}$.