Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Show that the nested intervals property fails for the rational numbers

  1. Mar 25, 2010 #1
    In the field of formal rational functions, construct a nest of closed, bounded intervals whose intersection is empty. (That is, show that the Nested Intervals property fails in this field)


    I know it has to involve radical 2 but because that is the only number we know is irrational but other than that I have no idea.
     
  2. jcsd
  3. Mar 25, 2010 #2
    Try 'centering' your intervals around an irrational (you should know more irrationals than that, but sqrt2 is still a good choice). For example, one interval can be [sqrt2 - 1/2, sqrt2 + 1/2]. Turn this into a nested sequence and consider what happens as the length of the intervals get very small.
     
  4. Mar 25, 2010 #3

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Is there a topology assumed for this field? If so, what is it? I'm not sure what an "interval" looks like in this field.
     
  5. Mar 25, 2010 #4
    I suspected he meant the set of rational numbers (since this is a common problem and he didn't know any other irrationals so odds are he doesn't know what a topology is).
     
  6. Mar 25, 2010 #5

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Good point. I would like to point out one subtlety that he might otherwise overlook, namely that [sqrt2 - 1/2, sqrt2 + 1/2] in the rational field does NOT contain the endpoints since sqrt2 +/- 1/2 is irrational. It is nonetheless a closed (AND an open) interval.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook