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Open Intervals

  1. Mar 15, 2013 #1
    Since we all agree that 1 = 0.999..., what does PF think the largest number in the open interval (0,1) is?
     
  2. jcsd
  3. Mar 15, 2013 #2

    micromass

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    What do you think the smallest number in the interval is?
     
  4. Mar 15, 2013 #3

    pwsnafu

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    Just because you can ask a question, doesn't mean there is an answer. (0,1) doesn't have a largest element. It's as simple as that.
     
  5. Mar 15, 2013 #4
    I always imagined that the smallest number less than one was the infinitesimal 0.9...., but that cannot be, because that equals one. The only other answer I could come come up with was that only terminating decimals of the form 0.999...9 would be in the interval, but that's no good either.

    I guess the smallest number would be of the form 0.000.....01 and it too would be non-terminating.
     
  6. Mar 15, 2013 #5

    jbunniii

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    But that number appears to terminate. Is there anything after the 1?

    You are overthinking this. We agree that (0,1) does not contain 0. So suppose it has a minimum, call it M. This is a positive number. Then M/2 is a smaller positive number. This is a contradiction, so there is no minimum after all.
     
  7. Mar 16, 2013 #6

    HallsofIvy

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    The open interval (0, 1) has NO largest or smallest member. You cannot talk about "the form 0.000.....01 and it too would be non-terminating". If is non-terminating, it does not have a last digit so you cannot assert that it is 1.
     
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