(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Proofs involving subsequences.

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