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Math Help - Infinite Sets and Cluster Points

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    Infinite Sets and Cluster Points

    How would you show that an infinite set \mathbb{N} has no cluster points...
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    Quote Originally Posted by frenchguy87 View Post
    How would you show that an infinite set \mathbb{N} has no cluster points...
    If n\in\mathbb{N} can the open set \left(n-\frac{1}{2},n+\frac{1}{2}\right) contain an point of \mathbb{N} other than n?

    If x\notin\mathbb{N} is there an open set containing x and no point of \mathbb{N}.
    Hint: \left( {\exists j \in \mathbb{Z}} \right)\left[ {j < x < j + 1} \right].
    Let \delta  = \min \left\{ {\left| {x - j} \right|,\left| {j + 1 - x} \right|} \right\}
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by frenchguy87 View Post
    How would you show that an infinite set \mathbb{N} has no cluster points...
    A little more generally, any set E such that E such that d(e,e')>\delta>0 for all e,e'\in E has no limit points. To see this let x be an arbitrary point and consider B_{\frac{\delta}{2}}(x). It either contains one or zero points of E. If the former we're done, if the latter take the open ball of one-half the distance from x to that point.
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