Does Every Subsequence of a Sequence Converge to the Same Limit?

In summary, the statement is asking to prove that a sequence {Xn} in R converges to a if and only if every subsequence of {Xn} also converges to a. The proof for one direction involves choosing a positive number e and showing that every subsequence of {Xn} is also within e of the limit a. The other direction may require showing that the sequence is Cauchy.
  • #1
mrroboto
35
0

Homework Statement



Suppose that {Xn} is a sequence in R. Prove that Xn converges to a if and only if every subsequence of Xn converges to a.

Homework Equations





The Attempt at a Solution



Let e>0, choose N in N st n >=N implies |Xn-a| <e. Since a subsequence, nk, is in N and n1<n2<n3..., then nk>=k for all k in N. So, k>= implies |Xnk-a|<e.

That's the first part, but I can't figure out how to start the proof the other way around. i.e. how do you prove that a convergent subsequence implies a convergent sequence?
 
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  • #2
It seems to me the sequence must be Cauchy for it to work.
 
  • #3
Any sequence is a subsequence of itself. So it's a tautology really.
 

FAQ: Does Every Subsequence of a Sequence Converge to the Same Limit?

1. What is subsequence convergence?

Subsequence convergence is a mathematical concept that describes the behavior of a sequence, or a list of numbers, as it approaches a specific value. More specifically, it refers to the idea that a subsequence, or a smaller list of numbers taken from the original sequence, will also approach the same value as the original sequence.

2. How is subsequence convergence different from convergence?

Subsequence convergence is a type of convergence that only applies to subsequences, whereas general convergence applies to the entire sequence. This means that a sequence can converge without all of its subsequences converging, but if a sequence has a subsequence that converges, then the entire sequence must also converge.

3. What is the importance of subsequence convergence in mathematics?

Subsequence convergence is an important concept in mathematics because it allows us to make predictions about the behavior of a sequence based on the behavior of its subsequences. This can help us understand the overall convergence or divergence of a sequence, and can also be useful in proving the convergence of more complex mathematical series.

4. How can we determine if a sequence has subsequence convergence?

In order for a sequence to have subsequence convergence, its subsequences must approach the same limit as the original sequence. This means that if we can show that all of the subsequences of a sequence converge to the same value, then we can conclude that the original sequence also converges. On the other hand, if a sequence has at least one subsequence that does not converge, then the sequence itself cannot have subsequence convergence.

5. Are there any real-world applications of subsequence convergence?

Yes, subsequence convergence has many real-world applications, particularly in fields such as physics and engineering. For example, it can be used to predict the behavior of a system over time, such as the decay of radioactive elements, or the convergence of a numerical method for solving a complex equation. It can also be applied in data analysis and prediction, such as in stock market trends or weather forecasting.

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