$ \begin{align*} \Pr[\text{bin } i \text{ has at least } k \text{ balls}] &\leqslant \left( \frac{e}{k} \right)^k = \left( \frac{e \ln \ln n}{3 \ln n} \right)^{\frac{3 \ln n}{\ln \ln n}} \\ &\leqslant \exp \left( \frac{3 \ln n}{\ln \ln n} (\ln \ln \ln n - \ln \ln n) \right) \\ &= \exp \left( - 3 \ln n + \frac{3 \ln n \cdot \ln \ln \ln n}{\ln \ln n} \right) \end{align*} $
When $n$ is large enough, $ \Pr[\text{bin } i \text{ has at least } k \text{ balls}] \leqslant \exp ( - 2 \ln n ) = \frac{1}{n^2}. $
This was found in this set of lecture notes.
Can anyone explain why the last step ("When n is large enough..") is true?