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Math Help - Subsequential Limits

  1. #1
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    Subsequential Limits

    Construct a sequence {sn} for which the subsequential limits are {-infinity, -2,1}

    Construct a sequence {sn} for which the set of subsequential limitsof the sequence are countable.

    Any help getting started would be great, I don't know where to begin.
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  2. #2
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    Re: Subsequential Limits

    Let me give you a few examples:
    To make a sequence with subsequential limits 0 and 1, how about:
    0, 1, 0, 1, 0, 1, 0, 1, ....
    or
    1/3, 2/3, 1/4, 3/4, 1/5, 4/5, ..., 1/n, (1-1/n), ...

    To make a sequence with subsequential limits 0 and 1 and -1, how about:

    a_n = \sin\left(\frac{\pi n}{2}\right), n \ge 0, which is 0, 1, 0, -1, 0, 1, 0, -1, ...

    To make a sequence with subsequential limits e and infinity, how about:

    a_n = \left(1+\frac{1}{n}\right)^n for n even and a_n = n for n odd.

    For the subsequential limit set being countable, I'd suggest trying to make it the positive integers - something easy.

    Can you blend the following infinite set of sequences into a *single* sequence?
    1, 1, 1, 1, ...
    2, 2, 2, 2, ...
    3, 3, 3, 3, ...
    4, 4, 4, 4, ...
    etc.
    (Hint: do you recall the method that's usually used to prove that a countable union of countable sets is a countable set?)
    Last edited by johnsomeone; October 8th 2012 at 02:00 PM.
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    Re: Subsequential Limits

    I'm afraid I don't really understand. The idea of subsequential limits makes sense, but for the first part I am having a lot of trouble seeing how to put in negative infinity. For part b, I'm afraid I still don't really understand.
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    Re: Subsequential Limits

    Quote Originally Posted by renolovexoxo View Post
    I'm afraid I don't really understand. The idea of subsequential limits makes sense...
    Since you're having trouble doing this, I'd have to disagree. What's going on hasn't really become clear to you yet.
    1) Did you look at my examples, and think about why they work?
    2) Can you do part a without the negative infinity? A sequence whose subsequential limits are -2 and 1?
    3) How about a sequence whose subsequential limits are -10, -2, and 1?
    4) Can you produce any simple ordinary sequence that goes to negative infinity? If so, then replace the terms of the sequence in #3 that are going to -10 with the terms of your sequence going to negative infinity, and you should have it.

    Quote Originally Posted by renolovexoxo View Post
    For part b, I'm afraid I still don't really understand.
    It's more important to worry about part a, since you can't run before you can walk.
    If you figure out part a, and think you really understand it, then maybe try to tackle part b. Maybe reread my first post's ideas about part b then.
    Until part a seems absurdly simple to you, there's no point in worrying about part b. Don't worry - it'll click eventually if you continue to think about and actively work on it.
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    Re: Subsequential Limits

    Quote Originally Posted by renolovexoxo View Post
    I am having a lot of trouble seeing how to put in negative infinity.
    s_n  = \left\{ {\begin{array}{rl}   { - n,} & {n \equiv _3 0}  \\   { - 2 + \frac{1}{n}.} & {n \equiv _3 1}  \\   {1 - \frac{1}{n},} & {n \equiv _3 2}  \\\end{array}} \right. where n \equiv _3 0 means \mod(n,3)=0

    So we have three sub-sequential limits: -\infty,~-2,~\&~1
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