Let us consider an operator proportional with a_\alpha^{\dagger} a_\beta and
\alpha=\beta . It acts on an n -particle state
resulting in
\begin{equation}
a_\alpha^{\dagger} a_\alpha |\alpha_1\alpha_2 \dots \alpha_{n}\rangle =
\begin{cases}
0 &\alpha \notin \{\alpha_i\} \\
\\
|\alpha_1\alpha_2 \dots \alpha_{n}\rangle & \alpha \in \{\alpha_i\}
\end{cases}
\tag{28}
\end{equation}
Summing over all possible one-particle states we arrive at
\begin{equation}
\left( \sum_\alpha a_\alpha^{\dagger} a_\alpha \right) |\alpha_1\alpha_2 \dots \alpha_{n}\rangle =
n |\alpha_1\alpha_2 \dots \alpha_{n}\rangle \tag{29}
\end{equation}