First note by Theorem 8.3 that if $\delta$ is fixed, then there exists $\delta^{\prime}({\rm v},\delta)$ such that if $|{\rm Ric}|\leq 3\delta^{\prime}$ and if $T^{\delta^{\prime}}_{0}=0$, then

$\displaystyle\big|\mathcal{V}^{\delta^{\prime}}_{1}(x)-\ln|\Gamma_{0}|\big|<\delta\,.$

(8.43)

By rescaling this inequality, we see that in the context of this lemma, the following holds. If $x\in B_{1}(p)$, $\alpha>\bar{\alpha}(v,\delta)$ and

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

(8.44)

then we have

$\displaystyle\big|\mathcal{V}^{\delta}_{r_{\alpha}}(x)-\ln|\Gamma_{\alpha}|\big|<\delta\,.$

(8.45)

In particular, for $x\in B_{1}(p)$, we can apply Lemma 8.5 to see that there exists a scale $\alpha\leq\bar{\alpha}(v,\delta)$ such that

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

(8.46)

and hence

$\displaystyle\big|\mathcal{V}^{\delta}_{r_{\alpha}}(x)-\ln|\Gamma_{\alpha}|\big|<\delta\,.$

(8.47)

However, if

$\displaystyle\mathcal{V}^{\delta}_{r_{\alpha}}(x)<\ln N-\delta\,,$

(8.48)

this implies $|\Gamma_{\alpha}|<N$, which completes the proof. ∎