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Math Help - Triangle Inequality

  1. #1
    MHF Contributor harish21's Avatar
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    Triangle Inequality

    Given y = (x+y)+(-x),
    I have to prove, using triangle inequality that ||x|-|y|| <= |x+y|

    ------------------------------------------------------------------


    This is what I did:

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

    so, |y| <= |x+y|+|-x|

    => |y|-|x| <= |x+y|

    How do I get to the conclusion?
    Last edited by harish21; February 9th 2010 at 03:00 PM.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    |y|=|(x+y)-x| \leq |y+x|+|x| so |y|-|x|\leq |x+y|. Reversing the roles of x,y we get |x|-|y| \leq |x+y|. So we have

    |y|-|x| \leq |x+y|

    and

    -(|y|-|x|) \leq |x+y|

    which means...
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  3. #3
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    For this we need to notice that:
    |a|\le |b| if and only if -|b|\le a \le |b|
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