Show that for \(A_{n}^{0}, A_{n}^{1} \subset X, n \in \mathbb{N}\), we have

\[\bigcup_{n \in \mathbb{N}}\left(A_{n}^{0} \cap A_{n}^{1}ight)=\bigcap_{i=(i(k))_{k \in \mathbb{N}} \in\{0,1\}^{\mathbb{N}}} \bigcup_{k \in \mathbb{N}} A_{k}^{i(k)} .\]