For $x\in B_{2}(p)$ fixed, we have

$\displaystyle{\rm Vol}(B_{1}(x))\geq C(n)^{-1}{\rm Vol}(B_{3}(x))\geq C^{-1}{\rm Vol}(B_{1}(p))\geq C^{-1}{\rm v}>0\,,$

(8.11)

and so,

$\displaystyle\mathcal{V}^{\delta}_{1}(x)\leq-\ln\Big(C^{-1}{\rm v}\Big)=C(n,{\rm v})\,.$

(8.12)

From the monotonicity of $\mathcal{V}^{\delta}_{r}(x)$, we have

	$\displaystyle C(n,{\rm v})-1\geq\mathcal{V}^{\delta}_{1}(x)-\mathcal{V}^{\delta}_{0}(x)$	$\displaystyle=\sum\Big(\mathcal{V}^{\delta}_{r_{\alpha}}(x)-\mathcal{V}^{\delta}_{r_{\alpha+1}}(x)\Big)$
		$\displaystyle=\sum\Big|\mathcal{V}^{\delta}_{r_{\alpha}}(x)-\mathcal{V}^{\delta}_{r_{\alpha+1}}(x)\Big|\,.$		(8.13)

In particular, there are at most $N=C(n,{\rm v})(\delta^{\prime})^{-1}$ elements $\alpha\in\mathds{N}$ such that

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

(8.14)

as claimed. ∎