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Math Help - Proof concerning absolute values and Triangle Inequality

  1. #1
    Junior Member hercules's Avatar
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    Proof concerning absolute values and Triangle Inequality

    Hey guys ...I haven't been online for six months. I need your help with the following question.


    Prove that \mid \mid a \mid - \mid b \mid \mid \leq \mid a-b \mid

    Thank You
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  2. #2
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    \begin{aligned}<br />
   \left| a-b \right|^{2}&=(a-b)^{2} \\ <br />
 & =a^{2}-2ab+b^{2} \\ <br />
  &\ge \left| a \right|^{2}-2\left| a \right|\left| b \right|+\left| b \right|^{2} \\ <br />
 & =\Big[ \left| a \right|-\left| b \right| \Big]^{2} \\ <br />
  \therefore\quad \left| a-b \right|&\ge \Big| \left| a \right|-\left| b \right| \Big|.\quad\blacksquare <br />
\end{aligned}
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by hercules View Post
    Hey guys ...I haven't been online for six months. I need your help with the following question.


    Prove that \mid \mid a \mid - \mid b \mid \mid \leq \mid a-b \mid

    Thank You
    I like Kriz's method! here's an alternate method:

    |a| = |(a - b) + b| \le |a - b| + |b| (The \triangle-inequality)

    Thus, we have |a - b| \ge |a| - |b|

    similarly, we can start with |b| and deduce that |a - b| \ge |b| - |a| \implies -|a - b| \le |a| - |b|

    putting them together we have -|a - b| \le |a| - |b| \le |a - b| \implies ||a| - |b|| \le |a - b| by the definition of absolute values
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  4. #4
    Math Engineering Student
    Krizalid's Avatar
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    Quote Originally Posted by Jhevon View Post

    (The \triangle-inequality)
    This can be proved in the same fashion as I did above.

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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Krizalid View Post
    This can be proved in the same fashion as I did above.

    yes, i know. that's the kind of proof i have for it

    i just used this method because he mentioned triangle inequality in the original post
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  6. #6
    Junior Member hercules's Avatar
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    Thank You both replies were really helpful.
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