- #1
quasar_4
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Homework Statement
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}
Homework Equations
Order Axioms and Field Axioms for the real numbers; infimum and supremum definitions; min and max definitions.
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?