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Thread: 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 $\displaystyle x_n \subseteq S$ 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 $\displaystyle x_n \color{red}\in\color{black} S$ of elements of S that converges to sup S
    Let $\displaystyle s=\sup S.$ Then $\displaystyle \forall\,n\in\mathbb N,$ $\displaystyle \exists\,x_n\in S$ such that $\displaystyle s\geqslant x_n>s-\frac1n$ since $\displaystyle s-\frac1n$ is not an upper bound for $\displaystyle S.$ The sequence $\displaystyle \left(x_n\right)_{n\,=\,1}^\infty$ converges to $\displaystyle s.$
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  3. #3
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    Quote Originally Posted by proscientia View Post
    Let $\displaystyle s=\sup S.$ Then $\displaystyle \forall\,n\in\mathbb N,$ $\displaystyle \exists\,x_n\in S$ such that $\displaystyle s\geqslant x_n>s-\frac1n$ since $\displaystyle s-\frac1n$ is not an upper bound for $\displaystyle S.$ The sequence $\displaystyle \left(x_n\right)_{n\,=\,1}^\infty$ converges to $\displaystyle s.$
    it's not belongs to !!!!
    both $\displaystyle 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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