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Math Help - Supremum of Confusion

  1. #1
    Banned
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    Supremum of Confusion

    I got from two different books a set B= \{r \in \mathbb{Q}: 0 \leq r \leq \sqrt{2}\}.

    One book says B does not have sup (B) and \sqrt{2}\not = sup(B) because \sqrt{2} \not \in B.

    The other book says sup(B) = \sqrt{2}. Although  \sqrt{2}\not \in B, there are rationals in the set arbitrarily close to \sqrt{2}.

    Who is right?
    The first book is a textbook, and the second is a supplement.
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  2. #2
    Super Member
    Joined
    Apr 2009
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    Depends on in which set are you looking at.

    If you are looking in the set of Q (rationals), B doesn't have a supernum in Q.

    If you are looking in the set of R (reals), B (and for that matter any bounded set) has a supernum in R. (This is the completeness property of the Reals.

    So both books are right. What you missed is the fact that one was talking of B as a subset of Q and other B as a subset of R
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