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: Proofs: Absolute Values and Inequalities

  1. Feb 8, 2012 #1
    1. The problem statement, all variables and given/known data

    I am wondering if the general approach to these proofs involving absolute values and inequalities is to do them case-wise? Is that the typical approach (unless pf course you see some 'trick')? For example, I have:

    Prove that if

    |x-xo| < ε/2 and Prove that if |y-yo| < ε/2,


    |(x+y)-(xo+yo)| < ε

    |(x-y)-(xo-yo)| < ε.

    It seems like there are way too many cases, but maybe that's just the way it is. For cases, I see:

    x=0, y=0

    0 ≤ x, y≤0

    oh jeesh .... forget listing them. I am now seeing that there are cases when xo is greater than or less than x (though I have a feeling that xo is supposed to be smaller than x and a positive number, but he (Spivak) did not specify...nor did he specify that ε > 0 though I think that it should be. Seems like these are gearing up for ε-δ proofs.

    Anyone have any thoughts on any of these points? This is problem 1.20 from Spivak's Calculus.
  2. jcsd
  3. Feb 8, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    |(x-x0)+(y-y0)|<=|x-x0|+|y-y0|. It's called the triangle inequality.
    Last edited: Feb 8, 2012
  4. Feb 8, 2012 #3
    Hi Dick :smile: I see that it works for one, I need to look at the (-) case a little more to see how it works for that one.

    But tell me: How many cases are there if I did not use the triangle inequality? I think I was going overboard in the OP. I need to look at when:

    1. x0+y0 < x+y < 0

    2. x0+y0 > x+y <0

    3. x0+y0 < x+y > 0

    4. x0+y0 > x+y > 0

    5. x0+y0 > 0 > x+y

    6. x0+y0 < 0 < x+y

    Am I missing anything? Sorry if it seems like I am beating a dead horse, but I never really got good at these and would like to fix that now.

    Thanks again.
  5. Feb 8, 2012 #4


    User Avatar
    Science Advisor
    Homework Helper

    You are fussing around about nothing. Just try to prove the triangle inequality using cases. Then call it a dead horse. Then you don't need to break every special case of it out into cases.
  6. Feb 8, 2012 #5
    Bah humbug to you too.
  7. Feb 8, 2012 #6


    User Avatar
    Science Advisor
    Homework Helper

    Sorry to rain on your parade of special cases. You'll thank me one day.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook