Put $\delta^{\prime}=\delta/3$. Then for $x\in B_{2}(p)$, there are at most $N({\rm v},\delta)=C({\rm v})\delta^{-1}$ scales $\alpha$ for which

$\displaystyle\Big|\mathcal{V}_{r_{\alpha}}(x)-\mathcal{V}_{r_{\alpha+1}}(x)\Big|>\frac{\delta}{3}\,.$

(8.16)

Hence, there are at most $3N$ elements $\alpha\in\mathds{N}$ such that

$\displaystyle\Big|\mathcal{V}_{r_{\beta}}(x)-\mathcal{V}_{r_{\beta+1}}(x)\Big|>\frac{\delta}{3}\,,$

(8.17)

for some $\beta\in\{\alpha-1,\alpha,\alpha+1\}$. Therefore, for all other $\alpha$, we must have

$\displaystyle\Big|\mathcal{V}_{4r_{\alpha}}(x)-\mathcal{V}_{r_{\alpha}/4}(x)\Big|<\delta\,,$

(8.18)

which proves the corollary. ∎