Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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)?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook