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Thread: Cuts

  1. #1
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    Cuts

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

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

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

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

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

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