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Math Help - Proof using Triangle inequality

  1. #1
    Member eXist's Avatar
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    Proof using Triangle inequality

    I have the problem:

    Prove:
    ||x| - |y|| \le |x - y|

    And I know I'm suppose to use the triangle inequality and the fact:
    x = x + y - y and y = y + x - x

    I tried starting with:

    |x - y| =>
    |(x - y + y) - (y - x + x)|

    But I don't know how to break this up using the triangle inequality:
    |x + y| \le |x| + |y|
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  2. #2
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    \left| x \right| = \left| {x - y + y} \right| \leqslant \left| {x - y} \right| + \left| { y} \right|\; \Rightarrow \;\left| x \right| - \left| y \right| \leqslant \left| {x - y} \right|

    Likewise: \left| y \right| = \left| {y - x + x} \right| \leqslant \left| {y - x} \right| + \left| x \right|\; \Rightarrow \;\left| y \right| - \left| x \right| \leqslant \left| {y - x} \right|

    Recall that |x-y|=|y-x|

    So putting the first two together we get.
     - \left| {x - y} \right| \leqslant \left| x \right| - \left| y \right| \leqslant \left| {x - y} \right|\; \Rightarrow \;\left| {\left| x \right| - \left| y \right|} \right| \leqslant \left| {x - y} \right|.
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  3. #3
    Member eXist's Avatar
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    I'm so upset with myself. I honestly tried that but never thought of moving |y| to the LHS. :/........

    Thanks so much . You're my hero.
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