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Math Help - infimum and supremum proof

  1. #1
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    infimum and supremum proof

    Prove, if S is a nonempty subset of real numbers that is bounded below then
    inf(S)= -sup{-s: s in S}.

    I have shown that the {-s:s in S} is bounded below and I let u = sup of that set but I do not know how to show -u = inf(S).
    Last edited by hayter221; October 5th 2008 at 02:19 PM.
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  2. #2
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    Suppose that \lambda  = \inf (S) then if follows that \left( {\forall z \in S} \right)\left[ {\lambda  \le z \Rightarrow \quad  - z \le  - \lambda } \right].
    That mean that -\lambda is an upper bound of the set <br />
-S.
    Say that \kappa  <  - \lambda then \lambda  <  - \kappa .
    By the definition of infimum \left( {\exists y \in S} \right)\left[ {\lambda  \le y < - \kappa } \right]
    That means that \left[ {\left( { - y \in  - S} \right) \wedge \left( {\kappa  <  - y \le  - \lambda } \right)} \right].
    This means that no number less that -\lambda is an upper bound of -S.
    Therefore  - \lambda  =  - \inf (S) = \sup ( - S).
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