I am trying to derive the fact that the function $F$ defined below is monotonically increasing. The only thing I can use is that the any member of the Cantor set has a ternary expansion involving only $0$'s and $2$'s. The function is defined as follows: Let $x\in[0,1]$ have ternary expansion $0.a_1a_2\cdots$. Define $N$ as the first index $n$ for which $a_n=1$ and set $N=\infty$ if none of the $a_n$ are $1$. Now let $F(x)=\sum_{n=1}^{N-1}\frac{a_n}{2^{n+1}}+\frac{1}{2^N}$.
I have shown that $F$ is constant while on a particular middle third removed in the construction of the Cantor set $C$ so I am really interested in showing the increasing nature on $C$. Essentially therefore I wish to show that given $x=\sum \frac{a_n}{3^n}<\sum\frac{b_n}{3^n}=y$ we have $\sum\frac{a_n}{2^{n+1}}<\sum\frac{b_n}{2^{n+1}}$ on $C$. How do I establish that?
Thanks