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

• MathematicalPhysicist
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.
MathematicalPhysicist
Gold Member
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?

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?
I think you are making too much of this. Since xn converges to a, given $\epsilon> 0$ there exist N1 such that if n> N1 then $|x_n-a|< \epsilon$. Since yn converges to a, given $\epsilon> 0$ there exist N2 such that if n> N1 then $|y_n-a|< \epsilon$.

What happens if you take N= max(N1, N2)?

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?

Yes, that is correct.

## 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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