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Math Help - Inequality

  1. #1
    Junior Member
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    Inequality

    is it true that

     ||z| - |w|| \leq |z + w| ??

    if so what is the proof?

    here is my workin so far.. please verify it thanks.

    We know  |z+w| \leq |z|+|w|
    let c = z - w, so  |c+w| \leq |c|+|w|

    Now z = c + w,
    so  |z| \leq |c|+|w|
     |z| \leq |z-w| + |w|
     |z-w| \geq |z| -|w|

    Now let d = - w,
    so  |z - d| \geq |z| - |d|
    subbing in -w for d we get,  |z + w| \geq |z| - |d|
    subbing in |w| for |d| since they are equal, we get  |z + w| \geq ||z| - |w||
    (i added an extra modulus bracket outside the right hand side at the end of the equation).

    End of proof.

    Is this correct please? please guide me if i am wrong?
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  2. #2
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    |x+y|\leq |x|+|y|.
    Thus,
    |x| = |x-y+y|\leq |x-y|+|y|\implies |x|-|y|\leq |x-y|
    And similarly,
    |y|-|x|\leq |x-y|
    Thus,
    ||x|-|y||\leq |x-y|.
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