Proving the Convergence of Sequence a_n to a: How to Show Unique Partial Limits?

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In summary, the sequence a_n, formed by alternating elements from the sequences x_n and y_n, converges to a due to the fact that both x_n and y_n converge to a. This is shown by taking the maximum of the indexes where the convergence of x_n and y_n is established, proving that a_n also converges to a.
  • #1
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i have that lim x_n=lim y_n=a
and we have the sequence (x1,y1,x2,y2,...)
i need to show that this sequence (let's call it a_n) converges to a.

well in order to prove it i know that if a_n has a unique partial limit then it converges to it to it, but how do i show here that it has a unique partial limit?
i mean if we take the subsequences in the even places or the odd places then obviously those subsequences converges to the same a, but i need to show this is true for every subsequence, how to do it?
thanks in advance.
 
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  • #2
loop quantum gravity said:
i have that lim x_n=lim y_n=a
and we have the sequence (x1,y1,x2,y2,...)
i need to show that this sequence (let's call it a_n) converges to a.

well in order to prove it i know that if a_n has a unique partial limit then it converges to it to it, but how do i show here that it has a unique partial limit?
i mean if we take the subsequences in the even places or the odd places then obviously those subsequences converges to the same a, but i need to show this is true for every subsequence, how to do it?
thanks in advance.
I think you are making too much of this. Since xn converges to a, given [itex]\epsilon> 0[/itex] there exist N1 such that if n> N1 then [itex]|x_n-a|< \epsilon[/itex]. Since yn converges to a, given [itex]\epsilon> 0[/itex] there exist N2 such that if n> N1 then [itex]|y_n-a|< \epsilon[/itex].

What happens if you take N= max(N1, N2)?
 
  • #3
yes i see your point.
if we take the max of the indexes then from there, we have that either way a_n equals x_n or y_n, and thus also a_n converges to a, cause from the maximum of the indexes both the inequalities are applied and thus also |a_n-a|<e, right?
 
  • #4
Yes, that is correct.
 

Related to Proving the Convergence of Sequence a_n to a: How to Show Unique Partial Limits?

1. What is a sequence question?

A sequence question is a type of question that requires a specific order or sequence of events to be answered. It is often used in science experiments and research to gather data and analyze results.

2. How are sequence questions used in scientific research?

Sequence questions are used in scientific research to collect and analyze data in a systematic and organized way. They help scientists identify patterns and relationships between variables and make accurate conclusions.

3. Can you give an example of a sequence question?

Sure, here's an example: "What is the sequence of events that occurs during photosynthesis?" This question requires a specific order of steps to be answered, such as "light energy is absorbed by chlorophyll, water is split, and carbon dioxide is converted into glucose."

4. How do sequence questions differ from other types of questions?

Sequence questions differ from other types of questions, such as open-ended or multiple-choice questions, because they specifically ask for a chronological or step-by-step response. They require a deeper level of understanding and analysis of a process or event.

5. Why are sequence questions important in scientific investigations?

Sequence questions are important in scientific investigations because they help scientists gather and interpret data in a structured and organized manner. They also allow for replication and verification of results, leading to more reliable and accurate conclusions.

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