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About definition of 'Bounded above' and 'Least Upper Bound Property'

  1. Mar 16, 2012 #1
    The definition of 'Bounded above' states that:

    If E⊂S and S is an ordered set, there exists a β∈S such that x≤β for all x∈E. Then E is bounded above.

    The 'Least Upper Bound Property' states that:
    If E⊂S, S be an ordered set, E≠Φ (empty set) and E is bounded above, then supE (Least Upper Bound) exists in S.

    My question is that why doesn't the definition of 'Bounded Above' include E≠Φ? Is there a problem when E=Φ? If not, then why does it matter when E=Φ for the 'Least Upper Bound Property'?
  2. jcsd
  3. Mar 16, 2012 #2


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    hi jwqwerty! :smile:
    Because Φ has no least upper bound (unless S is finite). :wink:
  4. Mar 16, 2012 #3


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    If we're working in [itex]\mathbb{R}[/itex], then we sometimes use the convention

    [tex]\sup \emptyset =-\infty[/tex]

    But we should be careful because [itex]-\infty[/itex] is NOT a real number. The above is not an equality of real numbers, but merely a notation.

    But don't ever use that notation in class unless your instructor uses it.
  5. Mar 16, 2012 #4


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    There is something missing in the statement of the existence of least upper bound. Example: Let S = set of all rational numbers and let E = set of all rational numbers < √2, then E is bounded, but supE is not in S.
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