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Where can I find a proof of the follwing fact?

If $$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$ for some function $u(x)$ satisfies $$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda w\left(u,x_{0},r\right)$$ for a fixed $0 < \lambda < 1$ and all sufficiently small values of $r$, then $u$ is Holder continuous.

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Just prove it yourself:

Take $r=1$. Then $$w(x_0,2^{-n})\leq \lambda^n w(x_0,1)=:C\lambda^n.$$ To estimate $|u(x_0)-u(y)|$, where $y$ is close to $x_0$, choose $n$ so that $|x_0-y|\in[2^{-n-1},2^{-n}].$ Then $$|u(x_0)-u(y)|\leq w(x_0,2^{-n})\leq C\lambda^{n}=C. 2^{-hn}\leq 2^hC_1|x_0-y|^h,$$ where $h=-\log\lambda/\log2$.

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