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Triangle Inequality Proof

  1. Sep 26, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove llxl-lyll≤lx-yl

    (The triangle inequality: la+bl≤lal+lbl)

    3. The attempt at a solution

    For the first part, I assumed lxl≥lyl:

    lxl=l(x-y)+yl

    Then, by Triangle Inequality

    l(x+y)+yl≤l(x-y)l+lyl

    So,
    lxl≤l(x-y)l+lyl
    Subtract lyl from both sides to get:

    lxl-lyl≤l(x-y)l.

    I'm not sure where to go from here. For, llxl-lyll≤lx-yl, don't I need to prove -l(x-y)l≤lxl-lyl≤l(x-y)l? How would I finish the proof?

    Thank you.
     
  2. jcsd
  3. Sep 26, 2012 #2

    micromass

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    Good. So you already proved

    [tex]|x|-|y|\leq |x-y| ~~~~~~~~~~~~~~(1)[/tex]

    Now, you need to prove the other inequality

    [tex]-|x-y|\leq |x| - |y|[/tex]

    You have done the hard work already. All you need to do now is to switch around x and y in (1).
     
  4. Sep 26, 2012 #3
    micromass- Okay, I did that:

    lyl=l(y-x)+xl

    And then,

    l(y-x)+xl≤l(y-x)l+lxl

    so, lyl≤l(y-x)l+lxl

    = lyl-lxl≤l(y-x)l

    What do I do from here? Would I factor out a negative from both sides?

    Like this:

    -(lxl-lyl)≤l-(x-y)l

    -(lxl-lyl)≤l-1ll(x-y)l

    And since l-1l=1

    -(lxl-lyl)≤l(x-y)l

    Finally,
    lxl-lyl≥l(x-y)l.

    Is this correct?
     
  5. Sep 26, 2012 #4

    micromass

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    No, that last step is incorrect. You have

    [tex]|y|-|x|\leq |y-x|[/tex]

    What if you multiply both sides by -1?? (Watch out, the inequality will reverse direction!!)
     
  6. Sep 26, 2012 #5

    micromass

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    Sorry, the last step was correct!

    So ignore my previous post.

    What is [itex]|-(x-y)|[/itex]?? Can you eliminate the -?
     
  7. Sep 26, 2012 #6
    I just edited my previous response, I think I got it (I hope)...
     
  8. Sep 26, 2012 #7
    One quick question, is it necessary for me to assume lxl is greater than or equal to lyl like I did in the beginning?
     
  9. Sep 26, 2012 #8

    micromass

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    That last step should read [itex]|x|-|y|\geq -|x-y|[/itex], but apart from that it's fine.
     
  10. Sep 26, 2012 #9

    micromass

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    Where do you assume that?? It doesn't seem necessary.
     
  11. Sep 26, 2012 #10
    I did in the beginning, but I guess it wasn't necessary.

    Thank you for all your help :)
     
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