Let $\delta>0$ be chosen with $\delta^{\prime}({\rm v},\delta)$ to be chosen later. Note that by Lemma 8.5, for each $x\in B_{1}(p)$ there exists $\alpha_{x}\leq\bar{\alpha}(v,\delta^{\prime})$ such that $T^{\delta^{\prime}}_{\alpha_{x}}(x)=0$. Consider the covering $\{B_{r_{\alpha_{x}}}(x)\}$ of $B_{1}(p)$, and choose an efficient subcovering $\{B_{r_{j}}(x^{\prime}_{j})\}_{1}^{N}$, where $r_{j}=r_{\alpha_{x^{\prime}_{j}}}$ and the balls in $\{B_{r_{j}/4}(x^{\prime}_{j})\}$ are disjoint. The usual doubling arguments imply that $N\leq N(v,\delta^{\prime})$.

By Theorem 8.3, if we are given $\epsilon>0$, then we can choose $\delta^{\prime}({\rm v},\epsilon,\delta)$ such that for each $x\in B_{\epsilon r_{j}}(x^{\prime}_{j})$ we have $T^{\delta}_{\alpha_{j}}(x)=0$, while for each $x\in A_{\epsilon r_{j},2r_{j}}(x_{j})$ we have $r_{h}(x)>\bar{r}(v,\epsilon)r_{j}\geq r_{0}(v,\epsilon,\delta^{\prime})$. Let $\Gamma_{j}$ be the group associated to $B_{r_{j}}(x^{\prime}_{j})$, and for each $x\in B_{\epsilon r_{j}}(x^{\prime}_{j})$ let $\beta_{j}(x)$ be the largest integer such that $V^{\delta}_{r_{\beta_{j}}/4}(x)\geq\ln|\Gamma_{j}|-\delta$. Let $\beta_{j}=\max\beta_{j}(x)$ with $x_{j}$ the corresponding point. Note that for $\epsilon(v,\delta)$ sufficiently small, we have $B_{2r_{\beta_{j}}}(x_{j})\subseteq B_{\epsilon r_{j}}(x^{\prime}_{j})$, and in particular, for every $x\in B_{2r_{\beta_{j}}}(x_{j})$

Consider the collection of balls $\{B_{r_{j}}(x_{j})\}$. Clearly, by construction, conditions (1) and (3) are satisfied. If $x\in B_{1}(p)\setminus\{B_{r_{j}}(x_{j})\}$ then since $\{B_{2r_{j}}(x_{j})\}$ cover $B_{1}(p)$ we have that for some $x_{j}$ that $x\in A_{r_{j},2r_{j}}(x_{j})$, which implies $r_{h}(x)\geq r_{0}(v,\delta)$, as claimed. ∎