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Math Help - a proof question..

  1. #1
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    a proof question..

    for "a" and "b" are real numbers prove that:
    ImageShack - Image Hosting :: 34860604zw7.gif
    Last edited by transgalactic; November 25th 2008 at 08:22 AM.
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  2. #2
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    \left| a \right| \leqslant \left| {a - b + b} \right| \leqslant \left| {a - b} \right| + \left| b \right|\; \Rightarrow \;\left| a \right| - \left| b \right| \leqslant \left| {a - b} \right|
    \begin{gathered}<br />
\left| b \right| \leqslant \left| {b - a + a} \right| \leqslant \left| {b - a} \right| + \left| a \right|\; \Rightarrow \;\left| b \right| - \left| a \right| \leqslant \left| {b - a} \right| = \left| {a - b} \right| \hfill \\<br />
- \left| {a - b} \right| \leqslant \left| a \right| - \left| b \right| \leqslant \left| {a - b} \right| \hfill \\ <br />
\end{gathered}
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  3. #3
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    how??

    first we have
    |a-b|>=||a|-|b||

    what did you do at the first step in order to transform it into

    http://img152.imageshack.us/img152/5374/52524050qr2.gif
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  4. #4
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    That is a simple application of the triangle inequality.
    In fact, the entire problem depends upon the triangle inequality.

    It also depends upon the general fact <br />
\left| a \right| \leqslant \left| b \right|\text{ if and only if } - \left| b \right| \leqslant a \leqslant \left| b \right|.
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  5. #5
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    can you tell me
    what operation you did in each step of this prove

    ??
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