Between any two points of the Cantor set there exists an interval contained entirely in the complement of C

Question

I have to prove that between any two points of the Cantor set there exists an interval contained entirely in the complement of the Cantor set.

Is the proof similar to that of showing that there exists between any two rational numbers there exist an irrational number? If it isn't I have no idea where to start.

Answer 1

Hint: Can you prove that there exists an interval between $0$ and $1$ contained entirely in the complement? After that, use the self-similarity properties of the Cantor set to prove it for any two points.

Answer 2

Since the Cantor set is constructed with an intersection that includes $[0,\frac{1}{3}] \cup [\frac{2}{3},1]$ and $(\frac{1}{3},\frac{2}{3}) \not\subset$ $[0,\frac{1}{3}] \cup [\frac{2}{3},1]$, $(\frac{1}{3},\frac{2}{3})$ is a subset of the complement of the Cantor set. This would contain an interval. This should be similar for every collection of intervals used in the intersection to construct the Cantor set.