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Math Help - zero content?

  1. #1
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    zero content?

    Let { x_{k}} be a converging sequence in R. Show that { x_{1}, x_{2}...} has zero content.
    I'm not even sure what the term zero content means?
    Thanks
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  2. #2
    Senior Member Tinyboss's Avatar
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    Content is a generalization of measure: http://en.wikipedia.org/wiki/Content_(measure_theory)

    You must be working with a particular content here, though, since the function that assigns 1 to a subset of R that contains 0, and 0 to every other subset is a content, and under this content any sequence that contains 0 has content 1.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by calculuskid1 View Post
    I'm not even sure what the term zero content means?

    You have to prove that for every \epsilon>0 there exists a finite family of intervals \{(a_i,b_i):i=1,\ldots,n\} which depends on \epsilon such that:

    (i) \{x_1,x_2,\ldots\}\subset \bigcup_{i=1}^{n}{(a_i,b_i) }

    (ii) \sum_{i=1}^{n}(b_i-a_i)<\epsilon

    Hint :

    As the sequence has a finite limit l, for every \epsilon>0 only a finite numbers of x_i do not belong to (l-\epsilon,l+\epsilon)


    Fernando Revilla
    Last edited by FernandoRevilla; January 22nd 2011 at 10:39 AM.
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