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  1. #1
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    sequence

    let's say that every subsequence of A = (a_n) has a subsequence that converges to 0. Show that lim A = 0.
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    If the lim X does not equal 0, then X either converges to a real number not equal to 0, or X diverges.

    So can I use a theorem that says that if X converges to a real number x, then any subsequence will also converge to x....since it was given that the subsequence converges to 0, this is a contradiction....?

    But I am having trouble with divergence. Also, in the question it is referring to a subsequence of a subsequence. Is this important? Or can I simply view a subsequence of a subsequence as a subsequence of X?
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    Quote Originally Posted by canadianmathite View Post
    If the lim X does not equal 0, then X either converges to a real number not equal to 0, or X diverges.

    So can I use a theorem that says that if X converges to a real number x, then any subsequence will also converge to x....since it was given that the subsequence converges to 0, this is a contradiction....?

    But I am having trouble with divergence. Also, in the question it is referring to a subsequence of a subsequence. Is this important? Or can I simply view a subsequence of a subsequence as a subsequence of X?
    question: is a sequence a subsequence of itself?
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    ...yes....
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    Quote Originally Posted by canadianmathite View Post
    ...yes....
    ok, so we can take this one step further and say that a sequence is itself a subsequence of one of its subsequences, correct?
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    Quote Originally Posted by Jhevon View Post
    ok, so we can take this one step further and say that a sequence is itself a subsequence of one of its subsequences, correct?

    yes, I'll agree with that....
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    Quote Originally Posted by canadianmathite View Post
    yes, I'll agree with that....
    ok, so if the sequence itself does not converge to zero, what can you say about a particular subsequence of one of its subsequences?
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    Quote Originally Posted by Jhevon View Post
    ok, so if the sequence itself does not converge to zero, what can you say about a particular subsequence of one of its subsequences?
    that it also does not converge to 0?
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    Quote Originally Posted by canadianmathite View Post
    that it also does not converge to 0?
    yes, and after re-reading your question, i realize that this doesn't help. haha. i should read more carefully

    let me reconsider in light of this new information

    (in case you are interested, the point i was getting at is that if we assume the subsequences of the subsequences converge to zero, yet, the sequence itself does not converge to zero, then we would have found a subsequence of the subsequence that does not converge to zero and hence arrive at a contradiction. however, the problem says each subsequence has "a" subsequence that converges to zero. not necessarily all subsequences of subsequences converge. so finding one that doesn't converge doesn't say anything, since you can argue there is some other subsequence that converge. with the "a" there, we only need one that converges, not all)

    i still think contradiction is the way to go though. we just have to work a bit harder
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    Quote Originally Posted by Jhevon View Post
    i still think contradiction is the way to go though. we just have to work a bit harder
    so am I still going to say that if X does not converge to 0, then:
    I. X converges to a real number not equal to 0, or
    II. X diverges

    and then do a proof by contradiction for both cases?
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    Quote Originally Posted by canadianmathite View Post
    so am I still going to say that if X does not converge to 0, then:
    I. X converges to a real number not equal to 0, or
    II. X diverges

    and then do a proof by contradiction for both cases?
    yes. either of these should not be that bad. there is a third case i am worried about though. what if the limit simply does not exist? as in, the sequence alternates, or jumps up and down between values.

    what about using limsup or liminf. have you done that in class? if we use those, then their limits will always exist, no need to worry about the bouncing up and down scenario
    Last edited by ThePerfectHacker; October 17th 2008 at 11:04 AM.
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    Kind of, but I don't know how that would apply since we don't know if sup or inf even exist....I am more confused than when I began
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    Quote Originally Posted by canadianmathite View Post
    Kind of, but I don't know how that would apply since we don't know if sup or inf even exist....I am more confused than when I began
    we do not need to know the supremum. it is enough to know that the limsup always exists. it either converges to a real number (particularly here, a non-zero one), or diverges to infinity or minus infinity.

    thus, you have those 3 cases to deal with. we cannot fulfill the condition that each subsequence has a subsequence converging to zero in any of those cases.
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    Quote Originally Posted by canadianmathite View Post
    I have been at this for awhile but am hesitant to make any conclusions. Help!
    assume \lim A \ne 0. then, in particular, \limsup A \ne 0. thus, we have 3 cases:

    (1) \limsup A = \infty
    (2) \limsup A = - \infty
    (3) \limsup A = L \ne 0, where L is some real number.

    cases (1) and (2) are similar. and the same argument works in general. thus, it suffices to deal with cases (1) and (3)

    case (1): \limsup A = \infty

    if \limsup A = \infty, then infinitely many terms of A become arbitrarily large. Thus, we can make a subsequence from such arbitrarily large elements. Say, a subsequence in which all the terms are greater than 1, for instance. then, clearly, no subsequence of that subsequence will converge to zero, which contradicts the defintion of A.

    case (3): \limsup A = L \ne 0

    if \limsup A = L, then infinitely many values of the sequence A get arbitrarily close to L. thus we can create a subsequence of terms "close" to L. In particular, terms that are closer to L than they are to 0. Then, for such a subsequence, no subsequence of that subsequence will converge to 0, which, again, contradicts how A was defined.

    thus, we have that \lim A = 0
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