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

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

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

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