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Math Help - triangle innequality question

  1. #1
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    triangle innequality question

    the question and prove:
    http://i33.tinypic.com/2gvk7k9.jpg

    they use the triangle innequality to prove itself
    ??
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  2. #2
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    That's not the triangle inequality that they proved. The proof is fine.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by transgalactic View Post
    the question and prove:
    http://i33.tinypic.com/2gvk7k9.jpg

    they use the triangle innequality to prove itself
    ??
    Problem: Prove that |x|-|y|\le|x-y|

    Proof:

    Lemma: |x+y|\le |x|+|y|

    Proof: Since \left||x|+|y|\right|=|x|+|y| the above is equivanlent to \sqrt{\left(x+y\right)^2}\le\sqrt{\left(|x|+|y|\ri  ght)^2}. Expanding both sides gives x^2+2xy+y^2\le|x|^2+2|x||y|+|y|^2. Noticing that |x|^2=|x|\cdot|x|=\left|x\cdot x\right|=x^2 we may rewrite this inequality as x^2+2xy+y^2\le x^2+2|x||y|+y^2 or xy\le |x||y| which is trivially true. \blacksquare

    Particularly the above tells us that |x|=\left|x-y+y\right|\le |x-y|+|y| or equivalently |x|-|y|\le |x-y|.


    Remark: Since we could have equally as well proven that |y|-|x|\le|x-y| we may actually strengthen this claim to \bigg||x|-|y|\bigg|\le|x-y|.
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  4. #4
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    your lemma is a half of the triangle innequality law
    and the thing i need to prove is its other half.
    so we using the law to prove the law itself

    ??
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  5. #5
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by transgalactic View Post
    your lemma is a half of the triangle innequality law
    and the thing i need to prove is its other half.
    so we using the law to prove the law itself

    ??
    I'm not sure what you mean. All the triangle inequality says is that \|x+y\|\leq\|x\|+\|y\|. You can (and should) use this to prove the given problem.
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  6. #6
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    ahhhh so what i need to prove is not the triangle innequality law
    in calc1 i was told other wise

    thanks
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