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Homework Help: Prove that F is complete

  1. Aug 22, 2010 #1
    1. The problem statement, all variables and given/known data
    Let F be an ordered field in which every strictly monotone increasing sequence bounded above converges. Prove that F is complete


    2. Relevant equations

    Definitions:

    Monotone Sequence property:
    Let F be an ordered field. We say that F has the monotone sequence property if every monotone increasing sequence bounded above converges.


    Completeness Property:
    An ordered field is said to be complete if it obeys the monotone sequence property

    3. The attempt at a solution

    Approach 1
    I'm not sure what exactly to prove. The question says that "strictly monotone increasing sequence bounded above converges" which is pretty much the monotone sequence property. And by the completeness property, F is complete . So what exactly am I supposed to do ? It seems trivial.

    Approach 2
    I could also get any strictly increasing sequence and extract an increasing subsequence which is bounded above and thus converges by monotone sequence property
     
    Last edited: Aug 22, 2010
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  3. Aug 22, 2010 #2

    Dick

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    The 'strictly monotone' sequences are a subset of the 'monotone' sequences. I think you have to take a monotone sequence and show it converges because it either contains a strictly monotone subsequence, or it's eventually constant.
     
  4. Aug 22, 2010 #3
    I see what you are trying to say but the problem with this approach is that I cannot readily find a way of doing this.

    Can I perhaps repeat some terms in the strictly monotone sequence to obtain a "regular" monotone sequence.

    Eg
    Let [tex]< a_{n}>[/tex] be a strictly monotone sequence then we define
    [tex]<a*_{n}>[/tex]
    where [tex] <a*_{n}> = <a_{0},a_{0},a_{1},a_{1}...>[/tex]
    where [tex] a_{0}, ... ,a_{n}[/tex] are terms of the strictly monotone sequence [tex] <a_{n}>[/tex].
    [tex] <a*_{n}>[/tex] is a "regular" monotone sequence and is increase and bounded above thus, it converges.

    Does this work ?



    PS I took a look at the back of the book I'm using and it says
    "From a( nontrivial) monotone sequence [tex] <x_{n}>[/tex], extract a subsequence that is strictly monotone ". WHAT ? . I understand the writers hint.
     
  5. Aug 22, 2010 #4

    Dick

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    I see what you are trying to do but it's easier if you follow the books hint. Start with a monotone sequence and delete terms to get a strictly monotone sequence. Then show since the strictly monotone sequence converges, so does the original.
     
  6. Aug 22, 2010 #5
    That seems good but considering that the chapter on cauchy sequences is further along I don't know any theorem to support my claim. I would have to show that the convergence, of a subsequence of a monotone increasing sequence, is enough to establish convergence of the sequence itself. I'm I over complicating things ?
     
  7. Aug 22, 2010 #6

    Dick

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    Overcomplicating. Just think epsilons and deltas. Use the definition of limit. You hardly need a theorem here. The terms you deleted are between the terms in the strictly monotone sequence.
     
  8. Aug 22, 2010 #7
    Perfect, I see it.

    Thanks a lot for the help.
     
  9. Aug 23, 2010 #8
    This can also seen to hold true as every Cauchy sequence is bounded & has a monotonic subsequence .
     
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