In Random Graphs 2001 (Bela Bollobas), p. 209, the proof for lemma 8.7, it says that $$\sum_{u=u_0-1}^{u_1}\sum_{w=1}^{\llcorner(\gamma - 1)u\lrcorner}(\log n)^w \Bigl(\frac{e}{u}\Bigr)^u\Bigl(\frac{eu}{w}\Bigr)^wn^{\gamma u^2/n} \leq \sum_{u=u_0-1}^{u_1}\gamma u(\log n)^{(\gamma - 1)u} \Bigl(\frac{e}{u}\Bigr)^ue^{(\gamma -1)u}n^{\gamma u^2/n}.$$ How this can be proved?
Actually, it seems to be equivalent to prove that $\sum_{w=1}^{\llcorner(\gamma - 1)u\lrcorner}(\frac{eu}{w})^w \leq \gamma ue^{(\gamma -1)u}$.
Also, how to prove that the formula following the formula above is $o(1)$ as in the book? Thank you in advance.
The theorem is as follows:
