Feller's Probability book goes from this step:
$$n^v \left(1-\frac{v}{n}\right)^{(v+r)}<v!S_{v}<n^{v}\left(1-\frac{v}{n}\right)^r $$
to this one:
$$\left(ne^{-\frac{v+r}{n-v}}\right)^{v}<v!S_{v}<\left(ne^{-\frac{r}{n}}\right)^{v}$$
I am clueless. Please help.
also,
$$S_{v}=\binom{n}{v}\left(1-\frac{v}{n}\right)^{r}$$
For the right inequality it suffices to show that $1-\frac{v}{n} \leq e^{-\frac{v}{n}}$ which is true.
For the left inequality it suffices to show that $e^{\frac{-v}{n-v}} \leq \frac{n-v}{n}$. Setting $x=\frac{-v}{n-v}$ the inequality is equivalent to $e^x \leq \frac{1}{1-x}$ for $x\leq0$, which is not hard to show.