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Math Help - Linear continua and the Real numbers

  1. #1
    Forum Admin topsquark's Avatar
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    Linear continua and the Real numbers

    Simple question, probably a simple answer, but I'm stuck.

    I'm trying to prove that the set of real numbers (in the usual order topology if you want specifics) is a linear continuum. I am having a problem with the least upper bound property. As I understand it, for the reals to have the lub property, any subset of the reals must have a lub. So what is the lub of the interval (1,+infinity)? It has no lub in the reals. But if it has no lub, then the set of real numbers doesn't have the lub property and thus is not a linear continuum! I've obviously gotten something wrong here, but I don't know what.

    Any help would be appreciated!

    Thanks.
    Dan
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  2. #2
    Super Member Rebesques's Avatar
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    What you need to do, is just to show that every bounded subset has a least upper bound.
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  3. #3
    TD!
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    Moreover, infinity isn't a real number.
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  4. #4
    Forum Admin topsquark's Avatar
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    Don't you hate it when you miss those tiny little words in the definition?

    Thanks for the help!

    Dan
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