1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook