Oscillation and Hölder continuity I am studying a proof of a theorem. And I have the following situation in the proof:
Consider $\Omega$ is a bounded open set of $\mathbb R^n$ and $u: \Omega \to \mathbb R$ is a function satisfying:
$$\operatorname{osc}_{B(x_0,R)} u \leq (1-\delta) \operatorname{osc}_{B(x_0,4R)},$$ for all ${B(x_0,R)} \subset \Omega$ for some $0<\delta <1$ ($\delta$ is independent of the open ball).
The book says: Iterating this inequality we have that $u$ is Hölder continuous.
Someone can help me understand the proof in the part of the "Iteration".
Thank you!
 A: I prefer to write the inequality as 
$$\operatorname{osc}_{B(x_0,4^{-1}R)} u \leq (1-\delta) \operatorname{osc}_{B(x_0,R)} \tag1$$
Iteration means that applying (1) with $B(x_0,R)$ replaced by $B(x_0,4^{-1}R)$, then replaced by $B(x_0,4^{-2}R)$, etc., and then chaining these inequalities together. Thus, 
$$\operatorname{osc}_{B(x_0,4^{-2}R)} u \leq (1-\delta) \operatorname{osc}_{B(x_0,4^{-1}R)} u \leq (1-\delta)^2 \operatorname{osc}_{B(x_0,R)} \tag2$$
and in general, 
$$\operatorname{osc}_{B(x_0,4^{-k}R)} u \leq (1-\delta)^k \operatorname{osc}_{B(x_0,R)} \tag3$$
Inequality (3) can be written in the form of the Hölder condition. Indeed, let $\alpha>0$ be the number such that $(1-\delta)=4^{-\alpha}$. Given $x$ near $x_0$, let $k$ be the integer such that $4^{-k-1}\le |x-x_0|< 4^{-k}R $. By (3), 
$$|u(x)-u(x_0)|\le (1-\delta)^k \operatorname{osc}_{B(x_0,R)} \le C |x-x_0|^\alpha \tag4$$
where $C=4^\alpha \operatorname{osc}_{B(x_0,R)}$ is independent of $x$. Thus, $u$ is  Hölder continuous.
