Hölder continuity of the Cantor function

The Cantor function (which I will define recursively below) is Hölder continuous and the exponent there is ln(2)/ln(3).

The function is defined as the limit function of a sequence of functions, where $ f_{n}$ looks as below: \begin{cases} f_{n-1}(3x/2) & 0 \leq x\leq 1/3\ 0.5 & 1/3\leq x\leq 2/3 \ 0.5 + f_{n-1}((3x-2)/2) & 2/3\leq x \leq 1 \end{cases}

I’ve proved that $ f_{n}$ converges uniformly to the Cantor function and that $ ||f_{n} – f_{n-1}||_{\infty}$ $ \leq$ $ \frac{1}{2^n}$ , but I have no idea where to proceed from there. Any inputs will be appreciated.