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Math Help - Cuts

  1. #1
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    Cuts

    The following is from "Foundation of Analysis," by Edmund Landau:

    Theorem 129:
    I) Let \xi and \eta be cuts. Then the set of all rational numbers which are representable in the form X+Y, where X is a lower number for \xi and Y is a lower number for \eta, is itself a cut.

    II) No number of this set can be written as a sum of an upper number for \xi and an upper number for \eta

    The author provided a 3-step proof, in which step 1, he proved the validity of statement (II). I understood this part.
    In step 2, he proved X+Y is a cut. I also understood this part, but I don't understand step 3.

    Step 3: Any given number of our set is the form X+Y where X and Y are lower numbers for \xi and \eta respectively. Using the third property of cuts (see * below) as applied to \xi, we can find a lower number
    x_1>x
    for \xi; then
    x_1+y>X+y
    So that there exists in our set a number which is >X+Y.

    Question: The author said, "we can find a lower number X_1>X for \xi."? Is X_1 the lower number? Is X_1 in \xi?

    *Definition 28: A set of rational numbers is called a cut if
    1) it contains a rational number, but does not contain all rational numbers;
    2) every rational number of the set is smaller than every rational number not belonging to the set;
    3) it does not contain a greatest rational number.
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  2. #2
    Member HappyJoe's Avatar
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    I'm not sure what the terms "lower number for a cut" means, but I'll try to answer you question anyway.

    You have that X is an element of the cut \xi. There _must_ be rational numbers in \xi larger than X, because otherwise X would be the largest number of \xi - and by part 3) of your definition, no cut contains a greatest rational number. So X_1 is just one of those rational numbers in \xi, which are larger than X.
    Last edited by HappyJoe; September 29th 2010 at 10:46 AM. Reason: Should have brushed up on greek letters.
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