Let $c_1>0$ and $0<a<1$ be real numbers. How to argue that there exists a constant $C>0$ such that $$a^{c_1\ln T}\leq T^{-C}$$ for all sufficiently large $T$, or even better, for all $T\in\mathbb{N}$?
The paper that I'm reading says that such $C$ exists but I'm not completely sure about it. Both sequences of the forms $a^{c_1\ln T}$ and $ T^{-C}$ converges to zero as $T\to \infty$. But to satisfy the inequality above I should pick $C$ so that $ T^{-C}$ converges to zero not faster than $a^{c_1\ln T}$ (otherwise the ratio of them would diverge).
My attempt
For all $T\in\mathbb{N}$, taking logs both sides \begin{align} a^{c_1\ln T}\leq T^{-C} &\iff c_1 \ln T \ln a\leq \ln 1-C\ln T\\ &\iff c_1\ln a\leq -C \end{align}
Thanks in advance.
Since $a\in(0,1)$, there must be some $C>0$ such that $a^{c_1}=e^{-C}$. So take $C=-c_1\log a$. Then
$$T^{-C}=T^{c_1\log a}=T^{\log a^{c_1}}=(e^{\log T})^{\log a^{c_1}}=(e^{\log a^{c_1}})^{\log T}=a^{c_1\log T}.$$