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Infinte number of terms from a sequence in a sub-interval

  1. Nov 12, 2012 #1
    I have across the following argument, which seems wrong to me, in a larger proof (Theorem 4 on page 9 of the document available at http://www.whitman.edu/mathematics/SeniorProjectArchive/2011/SeniorProject_JonathanWells.pdf). I would appreciate if someone can shed light on why this is true.

    The argument is that given a sequence $a_k$ of points in [a,b], we can say that a sub-interval of [a,b] exists such that it is smaller than some value $g<b-a$ and contains an infinite number of terms from $a_k$.

    I disagree with the above statement because lets say that the sequence $a_k$ always returns a constant value, say b. Then the above statement doesnt hold.
     
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  3. Nov 12, 2012 #2

    HallsofIvy

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    Yes, it does. It says "an infinite number of terms". It does NOT say "an infinite number of different terms". If your sequence were constant: an= b, then one those intervals will contain b and so contain an "infinite number" (in fact, all) of the terms of the sequence.
     
  4. Nov 12, 2012 #3
    Thanks for the clarification and I understand the point. But still I am not very convinced with the proof of Theorem 4 in the cited document above.

    The proof utilizes the above statement to conclude that two distinct values of the sequence ($a_k$), say $a_K$ and $a_K′$ would exist in a sub-interval of [a,b] such that $a_K−a_K′$ is greater than some value.

    Now, I think the above may not be possible and there is nothing in the original statement that guarantees that two distinct values of the sequence $(a_k)$ would fall in a subinterval of [a,b]. This may happen but we cannot say that it would happen for sure?

    Thanks again.
     
  5. Nov 12, 2012 #4
    Nah, the proof is fine except for one or two typos. It is not a_k - a_k' that is supposed to be large, but f(a_k)-f(a_k'). The point is this. The interval is partitioned into subintervals. Given the sequence a_k, an infinite number of these must fall into one of the bins (subintervals). That is true for any sequence in the original interval. Then he uses the fact that |f(a_k)| > k for each k to show that it is possible to select indices k and k' such that |f(a_k)|-|f(a_k')| is larger than n/(b-a).
     
  6. Nov 13, 2012 #5
    My understanding is that in the original theorem, we want to prove

    ∃y.∀x.a≤x∧x≤b⇒|fx|≤y


    If we contradict this statement, it becomes:

    ∀y.∃x.(a≤x∧x≤b)∧y<|fx|


    which can be further simplified to the following form:

    ∀k.(a≤(g m)∧(g m)≤b)∧m<|(f(g m))

    where m is a positive integer and (g_m) represents the sequence (a_k) in my original question.

    Now, the main concern that I have is that how can I mathematically prove that two distinct values of (g_m) would occur in a sub-interval of [a,b]. So basically, I dont understand the mathematical reasoning that allows us to reach the following statement that Vargo made from the last equation above.

    "Given the sequence a_k, an infinite number of these must fall into one of the bins (subintervals). That is true for any sequence in the original interval. "
     
  7. Nov 13, 2012 #6
    The statement is this. We are given an interval I which is partitioned into a finite number of subsets I_1, ...., I_M (which I refer to as bins). Let a_k be any sequence of numbers in I. Then there exists at least one bin, call it I_1, for which there exists an infinite set of indices [itex] K\subset \mathbb{N}[/itex] such that for all k in K, a_k belongs to I_1.

    That statement is true on the level of sets, meaning it has nothing to do with the fact that I is an interval and that I_1, ..., I_M are subintervals of a partition. And it has nothing to do with the nature of the sequence a_k. In other words, we can construct any sequence a_k and this fact will be true. In this case the author constructs the sequence a_k by using the hypothesis that f is unbounded on the interval and requiring that
    [itex] |f(a_k)|>k [/itex].

    Given that construction, let I_1 refer to a subinterval for which there an infinite number of indices k such that a_k belongs to I_1. Let K denote the set of indices for which a_k belongs to I_1. Choose k_1 in K. Now let B >0 be any positive number. K is an infinite subset of the positive integers, so it contains an element k_2 for which
    [itex] k_2 > B+ |f(a_{k_1})| [/itex]

    Then we conclude that a_{k_2} belongs to the same bin I_1 and
    [itex] |f(a_{k_2})| > k_2 > B+ |f(a_{k_1})[/itex]

    Now let B = n/(b-a).
     
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