The set of convergent subsequences

In summary, the conversation discusses finding examples of unbounded sequences with convergent subsequences within the interval [0,1] and to a specific number, such as 0. An example of such a sequence is shown using rational numbers and throwing in extra terms to make it unbounded. The conversation ends with a thank you to Dick for the help.
  • #1
zbr
11
0
Hello all,

(*) I have a question about convergent subsequences. Specifically I am looking for an example of a sequence that is unbounded but who has convergent subsequences in the interval [0,1].

A similar question would be to have an unbounded sequence, but who has a convergent subsequence to a specific number, let's say 0.

For this I would take the sequence:
a_n = -1,0,1,-2,0,2,-3,0,3,...,-n,0,n

Can I do a similar thing for (*)? i.e Can I take the following sequence:

a_n = -1, [0,1], 1, -2, [0,1], 2,...,-n, [0,1], n

My intuition tell me no since there will be infinite terms in each of the intervals [0,1].

Thanks in advance for any and all help!
 
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  • #2
zbr said:
Hello all,

(*) I have a question about convergent subsequences. Specifically I am looking for an example of a sequence that is unbounded but who has convergent subsequences in the interval [0,1].

A similar question would be to have an unbounded sequence, but who has a convergent subsequence to a specific number, let's say 0.

For this I would take the sequence:
a_n = -1,0,1,-2,0,2,-3,0,3,...,-n,0,n

Can I do a similar thing for (*)? i.e Can I take the following sequence:

a_n = -1, [0,1], 1, -2, [0,1], 2,...,-n, [0,1], n

My intuition tell me no since there will be infinite terms in each of the intervals [0,1].

Thanks in advance for any and all help!

You can arrange all of the rational number in [0,1] into a sequence, right? Now throw some extra numbers into make it unbounded, just like you did for the zero sequence.
 
  • #3
Ah, yes of coarse, set sequence 1, 1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, 1/16, ... would satisfy the sequence with convergent subsequences within the interval [0,1], then just throwing in divergent terms would satisfy the second requirement.

Thanks so much for your help Dick!
 

FAQ: The set of convergent subsequences

What is the definition of a convergent subsequence?

A convergent subsequence is a sequence of numbers within a larger sequence that approaches a specific limit as the number of terms increases. In other words, the numbers in a convergent subsequence get closer and closer to a single value as the sequence goes on.

How is a convergent subsequence different from a convergent sequence?

A convergent sequence is a sequence of numbers that approaches a single limit as the number of terms increases. In contrast, a convergent subsequence is a smaller sequence within a larger sequence that approaches a specific limit as the number of terms increases.

Can a sequence have multiple convergent subsequences?

Yes, a sequence can have multiple convergent subsequences. This occurs when there are multiple smaller sequences within the larger sequence that approach different limits as the number of terms increases.

How do you determine if a subsequence is convergent?

To determine if a subsequence is convergent, you must first identify the limit that the subsequence approaches as the number of terms increases. Then, you can use the definition of convergence to see if the subsequence satisfies the criteria.

What is the significance of convergent subsequences in mathematics?

Convergent subsequences are important in many areas of mathematics, particularly in analysis, calculus, and real analysis. They allow for a deeper understanding of the behavior of sequences and help in proving the convergence of larger sequences. They also have practical applications in fields such as physics, engineering, and computer science.

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