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Math Help - sequence question !

  1. #1
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    sequence question !

    Let S be a nonempty bounded above subset of R . prove that there is a sequence x_n \subseteq S of elements of S that converges to sup S
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  2. #2
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    Quote Originally Posted by flower3 View Post
    Let S be a nonempty bounded above subset of R . prove that there is a sequence x_n \color{red}\in\color{black} S of elements of S that converges to sup S
    Let s=\sup S. Then \forall\,n\in\mathbb N, \exists\,x_n\in S such that s\geqslant x_n>s-\frac1n since s-\frac1n is not an upper bound for S. The sequence \left(x_n\right)_{n\,=\,1}^\infty converges to s.
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  3. #3
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    Quote Originally Posted by proscientia View Post
    Let s=\sup S. Then \forall\,n\in\mathbb N, \exists\,x_n\in S such that s\geqslant x_n>s-\frac1n since s-\frac1n is not an upper bound for S. The sequence \left(x_n\right)_{n\,=\,1}^\infty converges to s.
    it's not belongs to !!!!
    both  x_n  \ and \  S are sets ,so the truth subsets
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  4. #4
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    [PHP][/PHP] You said "a sequence of elements of S". Did you mean "a sequence of subsets of S"? sup S is definitely a number. How does a sequence of sets converge to a number? I think you have misunderstood the question.
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