Let $G$ and $H$ be two groups generated by finite sets $\Sigma_G$ and $\Sigma_H$, and let $W=G \wr H$ be the wreath product of $G$ and $H$. Show that $W$ is finitely generated by $\Sigma_G \times \{1 \} \cup \{1 \} \times \Sigma_H$.
I cannot prove this. Please advice to me.
By definition, $G \wr H= \left( \bigoplus\limits_{h \in H} G_h \right) \rtimes H$ where $G_h$ is a copy of $G$. Therefore, $W= G \wr H$ is generated by $\Sigma_H \cup \{ \Sigma_{G_h} \mid h\in H \}$, and so by $\Sigma_H \cup \bigcup\limits_{h \in H} h \Sigma_G$ or by $\Sigma_H \cup \bigcup\limits_{h \in \Sigma_H} h \Sigma_G$. Finally, you can deduce that $W$ is generated by $\Sigma_H \cup \Sigma_G$.