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B Doubt regarding least upper bound?

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  1. Mar 2, 2016 #1
    I am using Spivak Calculus. I have a general doubt regarding the definition of least upper bound of sets.
    Let A be any set of real numbers and A is not a null set. Let S be the least upper bound of A.
    Then by definition "For every x belongs to A, x is lesser than or equal to S"

    Let M be an upper bound such that M is less than S and M does not belong to A(Assertion 1)

    Then, it leads to contradiction because A is bounded above by M and M is less than least upper bound S, which means an upper bound is less than least upper bound.
    So, we have either M=S or M>S. So, Assertion 1 is false.

    Then by converse of Assertion 1, if N is less than S, it may belong to A but can't be an upper bound or

    if N is less than S, it may not belong to A. (Assertion 2)

    Is my logic right?
     
  2. jcsd
  3. Mar 2, 2016 #2

    Svein

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    As long as your "or" and "and" are taken as logical operators and not English conjunctions, your loigic seems OK.
     
  4. Mar 2, 2016 #3

    Mark44

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    My take on this is that S really wasn't the least upper bound.

    Consider the set ##A = \{x \in \mathbb{R} | 0 \le x \le 1\}##; i.e., the open interval (0, 1).
    I claim that 1.5 is the lub for A. I later notice that 1 is not an element of A, and 1 < 1.5. This doesn't contradict anything, as all I have really done here is to mistakenly identify 1.5 as the lub.
     
  5. Mar 2, 2016 #4

    HallsofIvy

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    That's a property that every upper bound has, not just the least upper bound. To uniquely refer to the "least upper bound you must add "if M is an upper bound on A then [itex]S\le M[/itex]

    This is not an "assertion" because it is not a complete statement. You have defined M to have certain properties but have not said anything about it.
    And "let M be an upper bound such that M is less than S" violates the definition of least upper bound. Such an M does not exist.

    A number that is less than the least upper bound is cannot be an upper bound by definition of "least"!
     
  6. Mar 3, 2016 #5

    jbriggs444

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    That seems to have omited one word and should have read:

    A number that is less than the least upper bound is cannot be a least upper bound by definition of "least"!

    Edit: Doh, I misunderstood.
     
    Last edited: Mar 3, 2016
  7. Mar 3, 2016 #6

    HallsofIvy

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    No, it should be exactly what I said! Please read it again!
     
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