Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. We now assume that $\lim _ { n \rightarrow + \infty } \left( n ^ { \omega _ { 0 } } p _ { n } \right) = + \infty$.\\
Justify that for all natural integers $q$ and $r$ satisfying $1 \leq q \leq r$, we have :
$$\binom { r } { q } r ^ { - q } \geq \frac { 1 } { q ! } \left( 1 - \frac { q - 1 } { q } \right) ^ { q }$$
and deduce that for $k \in \llbracket 1 , s _ { 0 } \rrbracket$, we have $\Sigma _ { k } = \mathrm { o} \left( \left( \mathbf { E } \left( X _ { n } ^ { 0 } \right) \right)^{ 2 } \right)$ when $n$ tends to $+ \infty$.