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Is it true that ||z| - |w|| \leq |z + w| ?

  1. Oct 7, 2007 #1
    is it true that ||z| - |w|| \leq |z + w| ??

    is it true that

    [itex] ||z| - |w|| \leq |z + w| [/itex]??

    if so what is the proof?

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

    We know [itex] |z+w| \leq |z|+|w| [/itex]
    let c = z - w, so [itex] |c+w| \leq |c|+|w| [/itex]

    Now z = c + w,
    so [itex] |z| \leq |c|+|w| [/itex]
    [itex] |z| \leq |z-w| + |w| [/itex]
    [itex] |z-w| \geq |z| -|w| [/itex]

    Now let d = - w,
    so [itex] |z - d| \geq |z| - |d| [/itex]
    subbing in -w for d we get, [itex] |z + w| \geq |z| - |d| [/itex]
    subbing in |w| for |d| since they are equal, we get [itex] |z + w| \geq ||z| - |w|| [/itex]
    (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?
    Last edited by a moderator: Mar 10, 2013
  2. jcsd
  3. Oct 7, 2007 #2
    |z| =|z+w-w|<= |z+w|+|w|

    go from here
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