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Math Help - Bounding a sum of real functions with rationals

  1. #1
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    Bounding a sum of real functions with rationals

    I am stuck with this problem

    Show that for real functions f and g such that  f+g<x, there exists rationals r and s such that r+s<x, f<r and g<s

    Would using the Archimedean principle help?

    Let \alpha \in \mathbb{Q} and \alpha>0.
    By the Archimedean principle, there exists k \in \mathbb{Z} such that
    k\alpha \leq x < (k+1)\alpha

    Is this in the right direction?


    Thanks in advance
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  2. #2
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    Re: Bounding a sum of real functions with rationals

    Note that f<x-g. So, there exists rational r, such that f<r<x-g... And now think how to obtain second rational
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  3. #3
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    Re: Bounding a sum of real functions with rationals

    "Show that for real functions f and g such that  f+g<x, there exists rationals r and s such that r+s<x, f<r and g<s"

    let A =lubf, B=lubg

    If A and B are rational and belong to f and g respectiveley, you are finished.

    Otherwise you can pick rational numbers r and s greater than A and B respectiveley and arbitrarily close so that r+s < x
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