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
$$
\begin{equation}
\left( \sum_\alpha a_\alpha^{\dagger} a_\alpha \right) |\alpha_1\alpha_2 \dots \alpha_{n}\rangle =
n |\alpha_1\alpha_2 \dots \alpha_{n}\rangle \tag{29}
\end{equation}
$$