Sequence Proof (Am I missing something here?)

In summary, the conversation discusses a proof for a sequence of positive real numbers either containing a convergent subsequence or converging to positive infinity. The proof utilizes the bolzano-weirstrass theorem and considers both bounded and unbounded cases. However, the unbounded case may have a subsequence that converges to infinity and another subsequence that converges to a finite number.
  • #1
Newtime
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Sequence Proof (Am I missing something here??)

The reason I'm posting this is because I just took an exam and this was one of the questions, and it was so easy I feel like I may have been completely overlooking a complicating factor:

-{bn} is a sequence of positive real numbers. prove that either it contains a convergent subsequence or converges to positive infinity.

proof.

two case: either the sequence is bounded or unbounded. if it is bounded, apply the bolzano-weirstrass theorem to conclude that it contains a convergent subsequence. if unbounded, by definition, the sequence goes to positive infinity.

qed

On the exam I used better notation and wording but that's it essentially. so what's the consensus...is this valid?
 
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  • #2


lol nvm. Misread.
 
  • #3


Newtime said:
The reason I'm posting this is because I just took an exam and this was one of the questions, and it was so easy I feel like I may have been completely overlooking a complicating factor:

-{bn} is a sequence of positive real numbers. prove that either it contains a convergent subsequence or converges to positive infinity.

proof.

two case: either the sequence is bounded or unbounded. if it is bounded, apply the bolzano-weirstrass theorem to conclude that it contains a convergent subsequence. if unbounded, by definition, the sequence goes to positive infinity.

qed

On the exam I used better notation and wording but that's it essentially. so what's the consensus...is this valid?

The unbounded case is more complicated. It might have a subsequence which "converges" to infinity, as well as another subsequence which is convergent to a finite number.
 

What is sequence proof?

Sequence proof is a mathematical method used to prove the convergence or divergence of a sequence. It involves using logical arguments and mathematical principles to show that a sequence follows a certain pattern or trend.

Why is sequence proof important?

Sequence proof is important because it allows us to determine the behavior of a sequence, which is essential in many areas of mathematics and science. It helps us understand the limits of a sequence and make predictions about its future values.

What are the key steps in a sequence proof?

The key steps in a sequence proof include identifying the sequence and its general form, proving that it satisfies the necessary conditions for convergence or divergence, and then using mathematical techniques to show the desired result.

What are some common techniques used in sequence proof?

Some common techniques used in sequence proof include the comparison test, the ratio test, the root test, and the integral test. These tests help us compare a given sequence to a known sequence or function in order to determine its convergence or divergence.

What are some common mistakes to avoid in sequence proof?

Some common mistakes to avoid in sequence proof include failing to properly identify the sequence, using incorrect or incomplete mathematical techniques, and making logical errors in the proof. It is important to be thorough and precise when conducting a sequence proof to avoid these mistakes.

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