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Homework Help: Proofs involving subsequences.

  1. Mar 6, 2010 #1
    1. The problem statement, all variables and given/known data
    Which of the following sequences have a convergent subsequence? Why?

    (a) (-2)n

    (b) (5+(-1)^n)/(2+3n)

    (c) 2(-1)n

    2. Relevant equations
    Cauchy Sequence
    Bolzano-Weirstrass Theorem, etc.



    3. The attempt at a solution

    (a) The sequence I get is (-2,4,-8,16,-32,64...) We can get two subsequences, one comprised of (-2,-8,-32...) which diverges to -[tex]\infty[/tex], and (4,16,64...) which diverges to +[tex]\infty[/tex]. So there are no subsequences that converge?

    (b) The sequence I get here is something like (4/6, 7/8, 2/11, 9/14, 0, 11/20...) which seems to have no pattern. I don't know what to do here on out.

    (c) (1/2, 2, 1/2, 2, 1/2,...) So all odd integers n converges to 1/2, and all even integers n converges to 2.

    Here's my problem, other than being uncertain if I'm even doing this right: my professor wants rigorous proofs on everything we do in math, and I'm failing at writing adequate proofs. Where do I begin with a problem like this?
     
  2. jcsd
  3. Mar 6, 2010 #2
    What is the Bolzano Weierstrass Theorem? What conditions do you need to apply that theorem? Do any of your sequences fit that criteria?
     
  4. Mar 6, 2010 #3
    The Bolzano-Weierstrass Theorem simply states that "every bounded sequence has a convergent subsequence."

    It doesn't really seem to help here because the sequences themselves don't seem to be bounded.
     
  5. Mar 6, 2010 #4
    Really? The sequence that goes back and forth between 1/2 and 2 isn't bounded? What is the definition of a bounded sequence? Also for b), are you sure you wrote down your sequence correctly? As you have written it, the numerator goes back and forth between 4 and 6...
     
  6. Mar 6, 2010 #5
    snipez, thank you for correcting my obvious error when it comes to problem (b).

    And I know that the sequence from (c) is bounded, so the Bolzano-Weierstrass theorem will help me there.

    Do you have any suggestions on how to prove that these other subsequences converge (or don't)?
     
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