I am reading a book, *Randomized Algorithms* by Motwani.
In the Section 3.1 Occupancy problems, there is one step in the analysis that really puzzles me:

Let $k=\lceil (e \ln n) / (\ln\ln n) \rceil$,

$$(e/k)^k \; 1/(1-e/k) \le n^{-2}.$$

The book does not mention a single word about the above inequality.
Could anyone point out any clue for me? Thanks.

I try to simplify the inequality and get the following, but I have no idea about how to proceed.

$$\begin{align}
\ln \frac{(e/k)^k}{1-e/k} &= k - k \ln k - \ln(1-e/k)\\
&\le k - k\, \ln \frac{e \ln n}{\ln\ln n} - \ln(1-e/k)\\
&= - k \,\ln \frac{\ln n}{\ln\ln n} - \ln(1-e/k)\\
&...\\
&\le -\ln n^2
\end{align}$$