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Math Help - sequence - limit superior

  1. #1
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    sequence - limit superior

    I need to prove the following:
    Let Sn be a sequence in R. If s in R and for every e > 0, there exists n' in N s.t. Sn < s + e for all n >= n', prove that Lim superior Sn <= s

    Here is how I am approaching it:

    first, Sn - s < e -> s is the limit of Sn as n-> infinity.

    I also know that Lim Superior Sn = inf (sup[Sn:n>=k])

    So, maybe I could use both to say that Lim Sup (Sn) is either s or less than s

    What do you think?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by inthequestofproofs View Post
    I need to prove the following:
    Let Sn be a sequence in R. If s in R and for every e > 0, there exists n' in N s.t. Sn < s + e for all n >= n', prove that Lim superior Sn <= s

    Here is how I am approaching it:

    first, Sn - s < e -> s is the limit of Sn as n-> infinity.

    I also know that Lim Superior Sn = inf (sup[Sn:n>=k])

    So, maybe I could use both to say that Lim Sup (Sn) is either s or less than s

    What do you think?
    Let S be the set of all subsequential limits of \{S_n\} then \sup\text{ }S=\limsup\text{ }S_n try working with that.
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