Question on constructing a convergent sequence

In summary, the conversation discusses the possibility of constructing a sequence that converges to a given limit using elements from other convergent sequences. It is suggested that this can be done by setting the elements of the new sequence equal to the limits of the original sequences, but a proof is needed. The conversation also mentions that in the real numbers, every convergent sequence has a monotonic subsequence converging to the same limit.
  • #1
ScroogeMcDuck
2
0
Suppose for each given natural number n I have a convergent sequence [itex](y_i^{(n)})[/itex] (in a Banach space) which has a limit I'll call [itex]y_n[/itex] and suppose the sequence [itex](y_n)[/itex] converges to [itex]y[/itex].

Can I construct a sequence using elements (so not the limits themselves) of the sequences [itex](y_i^{(n)})[/itex] which converges to y? I would say [itex]z_n = y_n^{(n)}[/itex] would work, but I fail to prove this (my problem is making [itex]z_n^{(n)}[/itex] arbitrarily small for all n bigger than some natural number M)
 
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  • #2
in the reals, every convergent sequence has a monotonic subsequence converging to the same limit. But these could be increasing for some sequences and decreasing for others. So next you could argue there's either an infinite set of sequences with increasing subsequences or an infinite set with decreasing ones. Something along those lines might work for reals.
 

Related to Question on constructing a convergent sequence

1. What is a convergent sequence?

A convergent sequence is a sequence of numbers that approaches a specific limit as the index increases. This means that as the sequence continues, the numbers get closer and closer to a certain value.

2. How do you construct a convergent sequence?

To construct a convergent sequence, you need to start with a general term or rule that relates each term to the previous one. Then, you can plug in different values for the index to generate a sequence of numbers. Finally, you can check if the sequence approaches a specific limit as the index increases.

3. Can a convergent sequence have a limit of infinity?

No, a convergent sequence cannot have a limit of infinity. This would mean that the sequence does not approach a specific value and instead grows infinitely large, which is not a characteristic of a convergent sequence.

4. What is the difference between a convergent sequence and a divergent sequence?

A convergent sequence approaches a specific limit as the index increases, while a divergent sequence does not have a limit or approaches a limit of infinity.

5. How is a convergent sequence used in mathematics?

Convergent sequences are used in various mathematical applications, such as in calculus, where they are used to approximate values and solve problems related to limits and continuity. They are also used in analyzing the convergence of series and in various areas of physics and engineering.

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