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(Solved): Let (En); A Sequence Of Subsets Of X. Prove That A)x Limlower(En) = Limlower (En) And Xlimupper(En) ...

Let\thinspace (E_n)_{1}^{\infty}\thinspace a\thinspace sequence\thinspace of\thinspace subsets\thinspace of\thinspace X.\thinspace Prove\thinspace that\newline a) \chi\thinspace limlower(E_n)=limlower\thinspace (\chi E_n)\thinspace \thinspace and\thinspace \thinspace \thinspace \chi\thinspace limupper(E_n)=limupper(\chi E_n)\newline \newline b)(\chi E_n)_{1}^{\infty}\thinspace converges\thinspace if\thinspace and\thinspace only\thinspace if\thinspace limupper(E_n)=limlower(E_n)\thinspace so\newline limlower(\chi E_n)= limupper(\chi E_n)

Let (En); a sequence of subsets of X. Prove that a)x limlower(En) = limlower (En) and xlimupper(En) = limupper(En) b)(En) i converges if and only if limupper(En) = limlower(En) so limlower(En) = limupper(En)

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