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Math Help - Transitve relation help

  1. #1
    Senior Member Danneedshelp's Avatar
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    Transitve relation help

    I need to prove the intermediate value theorem using the following lemma:

    Lemma: Let be a transitive relation on the interval . If each has a neighborhood such that whenever and , then .

    So, the main goal is to actaully produce a transitive relation rho, but I am having no such luck. I'd assume \leq or < ought to work, but I am not sure what other conditions are needed to make the relation work in the proof.

    Also, does this prove the lemma given above?

    Let be arbitray and be a transitive relation on . By our hypothesis, we can find a neighborhood of such that whenever and . But, if we condsider the set it is clear that exists and because is compact . Since must share a transitive relation with and a transitive relation with , it must be that case that .

    Note: I have posted this question in the analysis thread, but I thought it might actaully fit better here, given it really comes down to using the proper relation.

    Thanks
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  2. #2
    MHF Contributor
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    Interesting use of relations in calculus.

    Quote Originally Posted by Danneedshelp View Post
    Also, does this prove the lemma given above?

    Let be arbitray and be a transitive relation on . By our hypothesis, we can find a neighborhood of such that whenever and .
    I don't understand the transition from here to the next statement below. It is strange that you abandon x that you have chosen, and in the following x appears only as a bound variable.
    But, if we condsider the set it is clear that exists and because is compact .
    To have one has to show that there are points in A that are arbitrarily close to b, i.e., that A does not, for example, begin with a end end with (a+b)/2. That, I think, is the main thing that one needs to show.

    I think this lemma is valid. Since for each point we have a neighborhood, you can choose a finite covering of [a,b]. Then, e.g., by induction on the number of neighborhoods in this covering it is easy to show that a\rho b.
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