Let \((X, \mathscr{A})\) be a measurable space and \(\left(\mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a strictly increasing sequence of \(\sigma\)-algebras, i.e. \(\mathscr{A}_{n} \subsetneq \mathscr{A}_{n+1}\). Show that \(\mathscr{A}_{\infty}:=\bigcup_{n \in \mathbb{N}} \mathscr{A}_{n}\) is never a \(\sigma\)-algebra.