Proofs involving subsequences.

[WaterPoloGoat]

Homework Statement

Which of the following sequences have a convergent subsequence? Why?

(a) (-2)ⁿ

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

(c) 2^(-1)n

Homework Equations

Cauchy Sequence
Bolzano-Weirstrass Theorem, etc.

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 -\infty, and (4,16,64...) which diverges to +\infty. 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?

 

[VeeEight]

What is the Bolzano Weierstrass Theorem? What conditions do you need to apply that theorem? Do any of your sequences fit that criteria?

 

[WaterPoloGoat]

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.

 

[snipez90]

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...

 

[WaterPoloGoat]

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)?