Prove that A** = A*, where A is a language over Σ*
Since $L \subseteq L^*$ for all $L$, we have $\mathcal{A}^* \subseteq \mathcal{A}^{**}$. In the other direction, suppose that $w \in \mathcal{A}^{**}$. Then there exists an integer $n \geq 0$ and words $x_1,\ldots,x_n \in \mathcal{A}^*$ such that $w = x_1 x_2 \ldots x_n$. Since $x_i \in \mathcal{A}^*$, there exists an integer $m_i$ such that $x_i \in \mathcal{A}^{m_i}$. Thus $w \in \mathcal{A}^{m_1 + \cdots + m_n} \subseteq \mathcal{A}^*$, and it follows that $\mathcal{A}^{**} \subseteq \mathcal{A}^*$.