Every bounded sequence is Cauchy?

In summary, a sequence can be bounded but not converge. However, the Bolzano-Weierstrass theorem states that there will always be a convergent subsequence. In the given example of {1,1,1,1,...}, the accumulation point is 1. This means that even though the sequence does not converge, it has subsequences that do converge to the accumulation point.
  • #1
CoachBryan
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I've been very confused with this proof, because if a sequence { 1, 1, 1, 1, ...} is convergent and bounded by 1, would this be considered to be a Cauchy sequence? I'm wondering if this has an accumulation point as well, by using the Bolzanno-Weirstrauss theorem.

I really appreciate the help guys. Thanks.
 
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  • #2
Cauchy is the same thing as convergent in the real numbers (or any complete metric space). A sequence can be bounded but not converge, like 0, 1, 0, 1, 0, 1...

However, the Bolzano-Weierstrass theorem says that there is a convergent subsequence, which in this case is easy to find directly: 0,0,0,0,0... or 1,1,1,1,1...

You get those by either skipping all the 1's or skipping all the 0's. The full sequence 0, 1, 0, 1, 0, 1... therefore has the accumulation points 0 and 1, since it has subsequences that converge to those points.

1, 1, 1, 1 has the accumulation point 1, the thing that it converges to.
 
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  • #3
homeomorphic said:
Cauchy is the same thing as convergent in the real numbers (or any complete metric space). A sequence can be bounded but not converge, like 0, 1, 0, 1, 0, 1...

However, the Bolzano-Weierstrass theorem says that there is a convergent subsequence, which in this case is easy to find directly: 0,0,0,0,0... or 1,1,1,1,1...

You get those by either skipping all the 1's or skipping all the 0's. The full sequence 0, 1, 0, 1, 0, 1... therefore has the accumulation points 0 and 1, since it has subsequences that converge to those points.

1, 1, 1, 1 has the accumulation point 1, the thing that it converges to.
Gotcha.

Thanks a lot, now it makes sense. So, not every bounded sequence is cauchy, but it's subsequences can be convergent (cauchy).
 

1. What is a bounded sequence?

A bounded sequence is a sequence of numbers where the values are all within a specific range or interval. This means that the values in the sequence do not exceed a certain upper bound and do not fall below a certain lower bound.

2. What is a Cauchy sequence?

A Cauchy sequence is a bounded sequence in which the distance between any two terms in the sequence approaches zero as the sequence progresses. In other words, the terms in the sequence get closer and closer together, and eventually become arbitrarily close.

3. What is the significance of a bounded sequence being Cauchy?

If a bounded sequence is Cauchy, it implies that the sequence is convergent, meaning it has a limit. This is significant because it allows us to make certain conclusions and predictions about the behavior of the sequence, even if we do not know the exact values of all the terms in the sequence.

4. How do you prove that a bounded sequence is Cauchy?

To prove that a bounded sequence is Cauchy, you can use the Cauchy criterion, which states that for any positive real number ε, there exists a positive integer N such that the distance between any two terms in the sequence after the Nth term is less than ε. This can be shown using the triangle inequality and the boundedness of the sequence.

5. Can a bounded sequence that is not Cauchy still be convergent?

No, a bounded sequence that is not Cauchy cannot be convergent. This is because the Cauchy criterion is a necessary condition for convergence. In other words, if a sequence is not Cauchy, then it cannot have a limit and therefore cannot be convergent.

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