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Math Help - Open Sets Question

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    Open Sets Question

    Let x \in O where O is open. Prove that if x_{n} \rightarrow x then all but a finite number of terms of (x_{n}) must be contained in O.
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    Moo
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    Hello,

    I suggest you go back to the topological definition of the convergence, and refer you to the fact that an open set is a neighbourhood.

    By the way, you sure this is a discrete math question, and not rather an analysis question ?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Janu42 View Post
    Let x \in O where O is open. Prove that if x_{n} \rightarrow x then all but a finite number of terms of (x_{n}) must be contained in O.
    As Moo said that is literally the definition of x_n\to x. If this is a metric space and that is not how it's defined, merely note that since x\in O there exists some \varepsilon>0 such that B_{\varepsilon}(x)\subseteq O. But, by definition there exists some N\in\mathbb{N} such that N\leqslant n\implies d(x_n,x)<\varepsilon\implies x\in B_{\varepsilon}(x)\implies x\in O
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