Let $C_k$ be the set obtained in the $k-$th stage of building the Cantor set, where
$$C_1=[0,\frac{1}{3}]\cup[\frac{1}{3},\frac{2}{3}]$$
$$C_2=[0,\frac{1}{9}]\cup[\frac{2}{9},\frac{1}{3}]\cup[\frac{2}{3},\frac{7}{9}]\cup[\frac{8}{9},1]$$
$$\dots$$
So we have
$$C_k=\bigcup_{j=1}^{2^k}I_{k,j}$$
where $I_{k,j}$ are pairwise disjoint closed intervals of length $3^{-k}$ in $[0,1]$, let
$$[0,1]\setminus C_k=\bigcup_{j=1}^{2^k-1}J_{k,j}$$
where $J_{k,j}$ are pairwise disjoint open intervals and each $J_{k,j}$ are located between intervals $I_{k,j},I_{k,j+1}$
define
$$F_k:[0,1]\to[0,1]$$
where
$$F_k(0)=0,F_k(1)=1,F_k(x)=\frac{j}{2^k}\quad x\in J_{k,j}$$
and $F_k$ is linear on each subinterval of $C_k$. We can show that $F_k$ converges to the Cantor function $F:[0,1]\to[0,1]$. Moreover, it can be shown that
$$|F_k(x)-F_k(y)|\leq\frac{3^k}{2^k}|x-y|$$
How can we show that $F$ is Hölder continuous of order $\log_32$ ?