Bounded sequence as convergent

In summary: But then why do you ask if it's clear?In summary, there are rules that state that not all bounded sequences must be convergent, as exemplified by the sequence Xn=(-1)^n. This can be proven by showing that there exist two different subsequences of Xn that converge to different numbers. Additionally, the negation of the proposition that x_n converges to x can be satisfied by exhibiting such a sequence. Finally, to prove that Xn=(-1)^n is not convergent, one can show that there exist two different subsequences that converge to different limits, such as the subsequence with n even converging to 1 and the subsequence with n odd converging to -1.
  • #1
electronic engineer
145
3
Some rule says that not all bounded sequence must be convergent sequence , one example is the sequence with general bound:
Xn=(-1)^n

could anyone help?!
thanks in advance!
 
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  • #2
What is it that you want help with? To show that x_n is divergent, it suffices to find two subsequences of x_n which converge to different numbers.
 
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  • #3
Ooh, erm, let me think... how about looking at the bloody definitions? Yes, I am tired, so delete as applicable, but quite frankly, I've seen enough of this for one day.
 
  • #4
electronic engineer said:
Some rule says that not all bounded sequence must be convergent sequence , one example is the sequence with general bound:
Xn=(-1)^n
could anyone help?!
thanks in advance!

if a sequence converges then it's bounded, but if it's bounded it might not converge as that example shows.
 
  • #5
fourier jr said:
if a sequence converges then it's bounded, but if it's bounded it might not converge as that example shows.

you're right, the question is how to prove that,could you help?!
 
  • #6
What is the negation of: x_n converges to x? Show that this is satisfied. Of course if x_n converges to x then all subsequences of x_n converge to x as well, maknig for an easy proof that it doesn't converge. If you're not used to working out the negations of propositions then say so.
 
  • #7
electronic engineer said:
you're right, the question is how to prove that,could you help?!
suppose that the sequence {a_n} converges. then a_n --> L for some finite L. now look at the definition of convergence & see that for ANY epsilon (yadda yadda)... for n>N but to make things easier just choose epsilon=1 & if n>N get 2 inequalities |a_n| < |a_n - L| + |L| < 1 + |L| (i think that's right, you check it). if n<N then |a_n| < something else.

so for any n>0, |a_n| < M where M=max{a certain set of numbers}. end of proof
 
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  • #8
electronic engineer said:
you're right, the question is how to prove that,could you help?!
Prove what? That there exist bounded sequences that are not convergent? You do that by exhibiting one, just as you did.

Prove that that particular sequence is not convergent? Just as Muzza said in the very first response to your post: show that there exist two different subsequences that converge to two different limits- in this case the subsequence with n even: 1, 1, 1, ... converges to 1, the subsequence with n odd: -1, -1, -1, ... converges to -1. Since arbitrarily far into the sequnce there exist numbers arbitrarily close to 1 and numbers arbitrarily close to -1, taking [itex]\epsilon= 1/3[/itex] will show that no number can be the limit of the entire sequence.
 
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What is a bounded sequence?

A bounded sequence is a sequence of numbers where there exists a finite number M such that all elements in the sequence are less than or equal to M. In other words, the values in the sequence do not grow infinitely large.

What does it mean for a bounded sequence to be convergent?

A bounded sequence is convergent if its elements approach a single, finite limit as the number of terms in the sequence increases. This means that the values in the sequence get closer and closer together as the sequence goes on.

How is the convergence of a bounded sequence determined?

The convergence of a bounded sequence can be determined by observing the behavior of its elements as the number of terms increases. If the values in the sequence approach a finite limit, the sequence is convergent.

What is the difference between a bounded sequence and a convergent sequence?

A bounded sequence is a sequence where the values do not grow infinitely large, while a convergent sequence is a bounded sequence where the values approach a single, finite limit as the number of terms increases. In other words, all convergent sequences are bounded, but not all bounded sequences are convergent.

Are all bounded sequences convergent?

No, not all bounded sequences are convergent. A bounded sequence may have elements that do not approach a single, finite limit as the number of terms increases, meaning it is not convergent. However, it is a necessary condition for a sequence to be convergent.

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