TITLE: $1/4$ is in the Cantor set?
QUESTION [14 upvotes]: I would like to know if $1/4$ is in the Cantor set, I tried a lot without success, I need some hint or help how to proceed in this case. Maybe there is some tool or trick I can use. 
Thanks a lot

REPLY [9 votes]: We use mathematical induction to show $\frac{1}{4}$ and $\frac{3}{4}$ are in $C_n,\, \forall n\in \mathbb{N}\cup \{0\}$.
Clearly, they are in $[0,1]$.
Suppose they are in $C_N$. We observe the structure of $[0,\frac{1}{3}]$ after $n+1$ cuts is similar to $[0,1]$ after $n$ cuts. Hence if $x \in C_N$, then $\frac{x}{3} \in C_{N+1}$. Since by assumption, $\frac{3}{4} \in C_{N}$, we have $\frac{1}{4} \in C_{N+1}$. Moreover, since the cut is symmetry on $[0,1]$, we know  $\frac{3}{4} \in C_{N+1}$.
Hence by induction, $\frac{1}{4}$ and $\frac{3}{4}$ are in $C_n,\, \forall n\in \mathbb{N}$, we have $\frac{1}{4}$ and $\frac{3}{4}$ are in $C$.