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Math Help - Greatest Integer Function

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    Greatest Integer Function

    Can you prove that for any real number x there exists as unique integer n such as,
    x-1\leq n\leq x
    thus, the function f(x)=[\x x\x ] is well-defined.
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  2. #2
    Super Member Rebesques's Avatar
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    Well... there certainly exists one such number, n(x); Or else, the naturals would be bounded in the reals, something the Archimedean Property denies.

    So there is one, at least. The set {n(x): n(x)-1< x <n(x)} must have a least element, by the Well-Ordering Property of the naturals. This least element, is exactly [x].

    Personally, I would not bother myself so much just take x, and kill its integer part!
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