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Math Help - Closed and bounded convergent sequences and subsets

  1. #1
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    Closed and bounded convergent sequences and subsets

    I've attached the problem. The last part of part a is: prove p is in F (in case it is hard to read)

    To me it intuitively makes sense that this p would be contained in F when it is closed and bounded, but I am having a ton of trouble seeing how to start the formal proof. Am I supposed to be using compactness?
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    Re: Closed and bounded convergent sequences and subsets

    Quote Originally Posted by renolovexoxo View Post
    I've attached the problem. The last part of part a is: prove p is in F (in case it is hard to read)
    To me it intuitively makes sense that this p would be contained in F when it is closed and bounded, but I am having a ton of trouble seeing how to start the formal proof. Am I supposed to be using compactness?
    The proof is trivial. Let \mathcal{O}=F^c, the complement of F.
    The complement of a closed set is an open set.
    If p\notin F then p\in\mathcal{O}.
    Thus p belongs to an open set that contains no point of F.
    That contradicts the given that p is a limit point of F~.
    Thanks from renolovexoxo
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