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Math Help - real numbers

  1. #1
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    real numbers

    I need help with this question,

    Does any open interval in R have a maximum? Explain your answer.

    Thomas
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  2. #2
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    Re: real numbers

    If it does, is it open? What say you?

    There must be a defintion of "Open Interval" sitting about somewhere. Why not have a good, close look at it?
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  3. #3
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    Re: real numbers

    i know it doesnt have any end points because you can always get a little bit more larger for example 0.1 then 0.11
    but i dont know to explain correctly
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  4. #4
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    Re: real numbers

    Quote Originally Posted by thomasboateng View Post
    i know it doesnt have any end points because you can always get a little bit more larger for example 0.1 then 0.11 but i dont know to explain correctly
    That may or may not be correct. It depends on which endpoint 0.1 is.
    However, this is the essential point: between any two real numbers there is a third number.
    If x\in (a,b) then a<x<b.
    Therefore \left( {\exists y \in (x,b)} \right)\left[ {x < y < b} \right].

    So can (a,b) have a maximal element?
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  5. #5
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    Re: real numbers

    Quote Originally Posted by thomasboateng View Post
    I need help with this question,

    Does any open interval in R have a maximum? Explain your answer.

    Thomas


    J=(a,b)


    \text{inf}\{ J \}=a

    \text{sup}\{ J \}=b

    a,b \not\in (a,b), hence by definition:

    \text{min}\{ J \}\neq a

    \text{max}\{ J \}\neq b
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  6. #6
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    Re: real numbers

    Indirect proof: Suppose the open interval, (a, b), does have a maximum, M. Since M is in (a, b), a< M< b. Let N= \frac{M+b}{2}.

    Prove:
    1) N< b.
    2) a< M< N
    so N is in (a, b)

    3) M< N, contradicting the hypothesis.

    Do you understand the difference between a "maximum" and a "supremum" (least upper bound)? That is crucial here.
    Last edited by Plato; July 31st 2011 at 05:42 AM. Reason: LaTeX fix
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