I know that the Cantor Set contains no segment of the form $\left(\frac{3k+1}{3^m}, \frac{3k+2}{3^m}\right)$ for any integers $k$ and $m$. If we can prove that every real segment contains a segment of that form, then certainly the Cantor Set contains no segment.
How do I show that for every $a, b \in \mathbb{R}$, there exist $k, m \in \mathbb{Z}$ such that $\left(\frac{3k+1}{3^m}, \frac{3k+2}{3^m}\right) \subset (a, b)?$
Presumably we can use the Archimedean Property of the real numbers, but I'm hazy on the details...