Proofs involving subsequences.

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In summary, the Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence. However, in this problem, the given sequences do not seem to fit the criteria for this theorem. The first sequence (-2)n has two subsequences that diverge to negative and positive infinity, respectively. The second sequence (5+(-1)^n)/(2+3n) was written incorrectly and should be (4/6, 7/8, 2/11, 9/14, 0, 11/20...), which does not seem to have a pattern and is not bounded. The third sequence 2(-1)n has a bounded subsequence that converges to 1/
  • #1
WaterPoloGoat
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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 -[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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  • #2
What is the Bolzano Weierstrass Theorem? What conditions do you need to apply that theorem? Do any of your sequences fit that criteria?
 
  • #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.
 
  • #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...
 
  • #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)?
 

1. What is a subsequence?

A subsequence is a sequence that can be derived from another sequence by removing some elements without changing the order of the remaining elements.

2. How are proofs involving subsequences useful?

Proofs involving subsequences are useful in mathematics and statistics as they can help establish properties and relationships of sequences, and can be used to prove the convergence or divergence of a sequence.

3. What are some common examples of proofs involving subsequences?

Some common examples of proofs involving subsequences include showing that a sequence is monotonic, bounded, or convergent.

4. What are some key concepts to understand when working with proofs involving subsequences?

Some key concepts to understand when working with proofs involving subsequences include the definition of a subsequence, the concept of convergence and divergence, and the use of theorems such as the Bolzano-Weierstrass theorem and the Cauchy criterion.

5. How can I improve my understanding of proofs involving subsequences?

To improve your understanding of proofs involving subsequences, it is important to practice solving problems and proofs, familiarize yourself with key concepts and theorems, and seek help from a teacher or tutor if needed.

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