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Math Help - Absolute value proof

  1. #1
    Senior Member OReilly's Avatar
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    Absolute value proof

    Can someone help me to understand this proof?

    I have to prove that:
    \left| {\left| x \right| - \left| y \right|} \right| \le \left| {x - y} \right|

    Proof from the book goes like this:
    1) let be x - y = z \Rightarrow x = y + z
    2) then is \left| x \right| = \left| {y + z} \right| \le \left| y \right| + \left| z \right|
    3) then \left| x \right| \le \left| y \right| + \left| z \right| \Rightarrow \left| z \right| \ge \left| x \right| - \left| y \right| \Rightarrow \left| {x - y} \right| \ge \left| x \right| - \left| y \right|
    4) same way we get \left| {y - x} \right| \ge \left| y \right| - \left| x \right|
    5) since also \left| {x - y} \right| = \left| {y - x} \right|{\rm{ and }}\left| {\left| x \right| - \left| y \right|} \right| = \left| {\left| y \right| - \left| x \right|} \right|
    6) we prove that \left| {x - y} \right| \ge \left| {\left| x \right| - \left| y \right|} \right|


    I understand everything except step 6).
    Can someone explain me that?
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  2. #2
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    Quote Originally Posted by OReilly
    6) we prove that \left| {x - y} \right| \ge \left| {\left| x \right| - \left| y \right|} \right|
    (I assume these are real numbers)

    There are two possibilities for here, on the right side.
    Either, |x|-|y|\geq 0 or |x|-|y|<0
    If the first case, then by the definition of absolute value,
    ||x|-|y||=|x|-|y| then by step 3 we know this is true.
    If the second case, thenby the definition of aboluste vale,
    ||x|-|y||=|y|-|x| then by step 4 we know it is true too.
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  3. #3
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    Thanks!
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