Prove that Cantor function is Hölder continuous
Using $\|F-F_k\|_\infty \le 2^{-k+1} \|F_0-F_1\|_\infty = 2^{-k}/3$ you get $$ |F(y)-F(x)| \le 2/3\cdot 2^{-k} + |F_k(y)-F_k(x)| \le (2/3 + 3^k |x-y|) 2^{-k}. $$ Now use $3^{-k-1} \le |x-y| < 3^{-k}$ for some $k$ to obtain $$ |F(y)-F(x)| < \frac{5}{3} 2^{-k} \le \frac{10}{3}|x-y|^{\log_3(2)} $$
Problem 4.12 in my lecture notes contains a hint for an alternate proof:
http://www.mat.univie.ac.at/~gerald/ftp/book-ra/ra.pdf