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Need someone to verify my logic!

  1. Sep 4, 2008 #1
    1. The problem statement, all variables and given/known data

    Find the supremum and infimum of the following sets and test whether these sets have a maximum or minimum:

    (a) { |x|/(1+|x|) s.t. x is in R}

    (b) { x/ (1+x) s.t. x > -1}

    2. Relevant equations

    Order Axioms and Field Axioms for the real numbers; infimum and supremum definitions; min and max definitions.

    3. The attempt at a solution

    I just don't see why these aren't effectively the same in terms of their supremums and infimums, min and max. Since |x|> 0 for nonzero x, |x| will always be positive, so this ratio will always be positive. Furthermore, it follows immediately that |x| < 1 + |x| for the same reason. Since the denominator is always bigger, the ratio is always less than 1. So I would argue that the supremum of (a) is 1. Now since we can include any x in R, the set also contains 0, and none of these can ever be less than 0. So 0 should be the infimum and serve as the minimum element of the set (since it is contained in the set).

    Now I can see how (b) would differ for all x in R, but we are now restricting x > -1, i.e.,
    x = 0, ... So it would seem that we have the exact same set - no negative x, the ratio is always less than 1, etc. So wouldn't this be exactly the same? What am I missing, if not?
  2. jcsd
  3. Sep 5, 2008 #2


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    Staff Emeritus
    Science Advisor

    I would think the are very obviously NOT the same. If x= -1.00001, what is x/(x+ 1)?
  4. Sep 5, 2008 #3
    oh yeah, I guess there are a bunch of negative reals between -1 and 0... wow that should have been obvious. I knew I was missing something simple. It always seems to be the case for me in analysis... :grumpy:

    since we could go "closer and closer" each time to -1, does that mean that this set doesn't have a lower bound? I mean, if you put in something like -0.99, then we get -0.99/(1-0.99) = -99, but then we could try x= -0.9999 and get something even smaller, and then we could keep going forever.
    I'm sorry if it's trivial.. I'm pretty dumb when it comes to this stuff.
    Last edited: Sep 5, 2008
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