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Finding a subsequence from a sequence that converges

  • Thread starter ppy
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  • #1
ppy
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Homework Statement


a real sequence (x[itex]_{n}[/itex]) is defined as follows: we take the elements in order (starting from
x0) to be

0, 1 , 0 , 1/10 , 2/10 ,... , 9/10, 1 0 , 1/100 ,2/100 ,..., 99/100 , 1 , 0 , 1/1000,...

So we take p for p = 0, 1, then p/10 for p = 0; .... 10, then p=100 for p = 0; ...., 100 and so on.
Which real numbers A have the property that some subsequence of (xn) converges to A?


Hi,

I am abit confused by what the question is asking do they want me to pick some values from the sequence and then this subsequence should converge to those values. But how can the sequence converge to all those values I have picked surely it can only converge to one of those values. I do not think I understand the question.

Help appreciated thanks.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618

Homework Statement


a real sequence (x[itex]_{n}[/itex]) is defined as follows: we take the elements in order (starting from
x0) to be

0, 1 , 0 , 1/10 , 2/10 ,... , 9/10, 1 0 , 1/100 ,2/100 ,..., 99/100 , 1 , 0 , 1/1000,...

So we take p for p = 0, 1, then p/10 for p = 0; .... 10, then p=100 for p = 0; ...., 100 and so on.
Which real numbers A have the property that some subsequence of (xn) converges to A?


Hi,

I am abit confused by what the question is asking do they want me to pick some values from the sequence and then this subsequence should converge to those values. But how can the sequence converge to all those values I have picked surely it can only converge to one of those values. I do not think I understand the question.

Help appreciated thanks.
The given sequence doesn't converge. But it has lots of different subsequences that converge. Can you think of some?
 
  • #3
ppy
64
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yes for example x4 to x13 which is the numbers from 1/10 to 1 converges to 1. is this all the question is asking? for me to write down different subsequences that converge to any number?
 
  • #4
Dick
Science Advisor
Homework Helper
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618
yes for example x4 to x13 which is the numbers from 1/10 to 1 converges to 1. is this all the question is asking? for me to write down different subsequences that converge to any number?
A subsequence of a sequence has to contain an infinite number of elements in the original sequence and in the same order. So, no, that's not one. Try again!
 
  • #5
ppy
64
0
I don't understand how u can find a subsequence that converges by keeping the terms in the same order because the terms will get larger then smaller then larger etc. so surely they are not converging to anything
 
  • #6
Dick
Science Advisor
Homework Helper
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I don't understand how u can find a subsequence that converges by keeping the terms in the same order because the terms will get larger then smaller then larger etc. so surely they are not converging to anything
Pick all of the terms that are 0. There are an infinite number of them, so that gives you the subsequence {0,0,0,0,0,...}. Which clearly converges to 0. Now find some more subsequences that converge.
 

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