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!

Homework Help: 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:


    Then, by Triangle Inequality


    Subtract lyl from both sides to get:


    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
    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:


    And then,


    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:



    And since l-1l=1



    Is this correct?
  5. Sep 26, 2012 #4
    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
    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
    That last step should read [itex]|x|-|y|\geq -|x-y|[/itex], but apart from that it's fine.
  10. Sep 26, 2012 #9
    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 :)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook