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Math Help - implication of a set of inequalities

  1. #1
    Per
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    implication of a set of inequalities

    I have had problems trying to prove that one inequality does not imply another inequality. Let ε, λ, X and Y be scalars where
    ε ≥ 0
    λ > 0
    -∞ < X < +∞
    -∞ < Y < +∞

    We have the following two inequalities:

    INEQUALITY 1:

    X Y ≥ ε λ

    INEQUALITY 2:

    if ε = 0 then X Y ≥ 0
    if ε > 0 then X Y > 0

    I want to show that INEQUALITY 1 implies INEQUALITY 2, but that the converse is not true. That is,
    i) If INEQUALITY 1 is satisfied then INEQUALITY 2 must be satisfied
    ii) If INEQUALITY 2 is satisfied then INEQUALITY 1 need not be satisfied

    While i) is easy to prove, I have gotten stuck on trying to prove ii). The thing is that I need to show it in the general case, and not with a specific example using made up numbers.

    I would be very grateful for all help I can get!!
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  2. #2
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    Quote Originally Posted by Per View Post
    I have had problems trying to prove that one inequality does not imply another inequality. Let ε, λ, X and Y be scalars where
    ε ≥ 0
    λ > 0
    -∞ < X < +∞
    -∞ < Y < +∞

    We have the following two inequalities:

    INEQUALITY 1:

    X Y ≥ ε λ

    INEQUALITY 2:

    if ε = 0 then X Y ≥ 0
    if ε > 0 then X Y > 0

    I want to show that INEQUALITY 1 implies INEQUALITY 2, but that the converse is not true. That is,
    i) If INEQUALITY 1 is satisfied then INEQUALITY 2 must be satisfied
    ii) If INEQUALITY 2 is satisfied then INEQUALITY 1 need not be satisfied

    While i) is easy to prove, I have gotten stuck on trying to prove ii). The thing is that I need to show it in the general case, and not with a specific example using made up numbers.

    Not at all! To show something general is not true it is enough ONE single counterexample...for example, X=2\,,\,Y=1\,,\,\epsilon=3\,,\,\lambda=45

    Tonio


    I would be very grateful for all help I can get!!
    .
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