I was reading this paper about Brun's theorem proof:
https://wiki.math.ntnu.no/_media/ma3001/2014v/analytisktallteori/tjerand_silde_project_ma3001.pdf
At page 6 it says (I write it without a typo): $$\sum_{f=m+1}^r{(-1)^{f-1}\cdot s_f} \leq s_{m+1} \leq \frac{s_1^{m+1}}{(m+1)!} \leq \left(\frac{es_1}{m+1}\right)^{m+1} \leq \left(\frac{1}{e}\right)^{m+1} \leq e^{-e^2 s_1} \leq e^{-s_1}$$
where $m\leq r$ is an even integer, $p_1,...,p_r$ are all the odd primes not exceeding a fixed value $y$ and $$s_f:=\sum_{1\leq i_1<\cdots<i_f\leq r}{\frac{2^f}{p_{i_1}\dots p_{i_f}}}$$
The first inequality is true by simmetry of binomial coefficient.
The second inequality is true by a previous result: $$s_f \leq \frac{s_1^f}{f!}$$
The third inequality is true by Stirling: $$n! \geq \left(\frac{n}{e}\right)^n\sqrt{2\pi n} \geq \left(\frac{n}{e}\right)^n \qquad \forall n\geq 1$$
The fourth and fifth inequalities are true by the condition $e^2 s_1 < m+1$.
The sixth inequality is trivially true.
The only thing I cannot understand is where the author use the other condition: $m+1 < 9s_1$?
I tried to figure it out with many examples, but I really don't need that condition. However he uses it in the next chapter...