Let $K$ the Cantor set. I already proved the following properties
Properties.\begin{equation} \begin{split} (1)&\quad|K|=|\mathbb{R}|\\ (2)&\quad\lambda(K)=0,\text{where}\;\lambda\;\text{is the Lebesgue measure};\\ (3)&\quad\text{If}\;E\in 2^{K}\Rightarrow E\;\text{is Lebesgue measurable;} \end{split} \end{equation}
I want to prove that $K$ does not contain intervals.
Now, if $I=[a,b]\subseteq K$ is an interval, $a\ne b$, then for $(3)$ it is measurable and $\lambda(I)>0$, absurd. Therefore, $K$ does not contain intervals.
Question. How can I show that $K$ contains no intervals using only the method by which it is constructed, ie without using the monotony of the measure?
Thanks!