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}}$ ?