It suffices to verify each condition for $\mathscr{F}$ in Lemma 6.1. Proposition 6.8 and Proposition 6.4 show that $\mathscr{F}$ satisfies Item (1) and Item (2a). In our context, $C_{L}>0$ and $C_{N}>0$ are uniform constants. $r_{0}>0$ can be chosen as any fixed constant in $(0,\frac{1}{2C_{L}C_{N}})$. To verify Item (2b) in Lemma 6.1, we just need to use (6.21). In fact, we have assumed $\nu+\alpha<0$, then

as $T$ is sufficiently large. This completes the proof.

∎