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Math Help - Sequence within a non-empty, bounded above set

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    Sequence within a non-empty, bounded above set

    Let S be a non-empty subset of R (real numbers) that is bounded above. Show that there exists a sequence (xn, n is a natural number), contained in S (that is, xn is an element of S for all n in the set of natural numbers) and which is convergent with limit equal to sup S.

    Any help would be greatly appreciated
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    Re: Sequence within a non-empty, bounded above set

    Quote Originally Posted by sakuraxkisu View Post
    Let S be a non-empty subset of R (real numbers) that is bounded above. Show that there exists a sequence (xn, n is a natural number), contained in S (that is, xn is an element of S for all n in the set of natural numbers) and which is convergent with limit equal to sup S.

    Suppose that \sigma=\sup(S). Two cases:
    1) \sigma\in S, what is a constant sequence that works?

    2) \sigma \notin S and \exists s_1\in S such that s<\sigma. You explain why!

    If n>1 then \exists s_n\in S such that \sigma-\tfrac{1}{n}\le s_n<\sigma. You explain why!
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