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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,

    then

    |(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

    Dick

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    |(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

    Dick

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    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

    Dick

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    Sorry to rain on your parade of special cases. You'll thank me one day.
     
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