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  • 1 Post By romsek

Math Help - Question

  1. #1
    Member
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    Question

    I didn't know where to ask this question because it's really simple, I uploaded a picture.

    It tries to show why the intersection of the all sets from [s,2] were s is defined to be 0<s<1 and it's clear to me that the answer is [1,2]

    However when he tries to prove why numbers less than 1 can't be in the intersection he assumes x<1 and then chooses s to be the mid point between 1 and x. and it doesn't satisfy the condition, but I don't get how he can just chose s t be the midpoint of x and 1. if x<1 then s<x<1<2 , you can't just choose s to be the midpoint between 1 and x ...

    It's like saying x is less than 2 and then choosing x to be 3 and say it's contradiction. Can someone explain the logic here? Clearly I am not understanding the thing he tried to do.
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  2. #2
    MHF Contributor
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    Re: Question

    He's trying to show that for any x you might choose less than one, there exists an s such that x < s < 1

    That should be pretty clear regardless of whatever particular s he might choose by the density of the real numbers. He just happens to pick s as the midpoint.

    So there exists an s such that x is not in [s,2] and thus x cannot be in the intersection of all such intervals.
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