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Math Help - Real analysis proof

  1. #1
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    Real analysis proof

    let 0<x show that there is a unique m in N such that m-1 < or = x < m.
    the book says to consider n in N : x < n and that N is well ordered
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by mtlchris View Post
    let 0<x show that there is a unique m in N such that m-1 < or = x < m.
    the book says to consider n in N : x < n and that N is well ordered
    For x\in\mathbb{R}, consider the set S_x=\{n\in\mathbb{N}:n>x\}. By the well ordering principle, this set has a unique smallest element. Call this element m.

    x<m because m\in S_x. m-1\leq x because if it isn't, then m-1\in S_x, contradicting the fact that m=\min(S_x).
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