# Prove that an iterative estimate implies Holder continuity

Let $$u$$, $$w$$ be nonnegative continuous functions such that $$\frac{u}{w}$$ is bounded on $$B_{2^{-1}}$$. Why the inequality $$a_k \le \frac{u}{w} \le b_k \quad \text{ on B_{2^{-k}}} , \qquad b_k - a_k < \delta^k,$$ ($$\delta < 1$$) for every $$k \in \mathbb{N}$$ implies that $$\frac{u}{w}$$ is Holder continuous on $$B_{2^{-1}}$$?