Is a Cauchy Sequence Always Convergent?

In summary, a Cauchy sequence is a sequence of real numbers that converges to a specific number as m and n go to infinity.
  • #1
soopo
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Homework Statement


What does it mean when a sequence is Cauchy?
 
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  • #2
A sequence of real numbers is a "Cauchy sequence" if and only if |an- am| goes to 0 as m and n go to infinity independently: given [itex]\epsilon> 0[/itex] there exist N such that if m and n are both > N, then [itex]|a_n- a_m|< \epsilon[/itex].

The advantage of working with Cauchy sequences is that (1) even if our sequence is of points in some abstract space, the "distance between points", here |p- q|, is a real number so we are now working with sequences of real numbers and (2) we know what we want the sequence to converge to.

Of course, for that to be useful, we have to know that the "Cauchy Criterion", that every Cauchy sequence converges, holds in our space- that our space is complete. That has to be proven separately. For exampe the set of real numbers is complete but the set of rational numbers is not. The sequence 3, 3.14, 3.141, 3.1415, 3.14159, ..., where each number contains one more digit in the decimal expansion of [itex]\pi[/itex] is a sequence of rational numbers (each number is a terminating decimal) and a Cauchy sequence (if m,n> N, am and an are identical for at least the first N decimal places so |am- an|< 10-N which goes to 0 as N goes to infinity) but does not converge to any rational number.
 
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  • #3
HallsofIvy said:
A sequence of real numbers is a "Cauchy sequence" if and only if |an- am| goes to 0 as m and n go to 0 independently: given [itex]\epsilon> 0[/itex] there exist N such that if m and n are both > N, then [itex]|a_n- a_m|< \epsilon[/itex].

So Cauchy sequence occurs when
If [tex]\forall \epsilon > 0 \exists N, m > N[/tex] and [tex]n > N[/tex], then
[tex] |a_{n} - a_{m}| < \epsilon.[/tex]
 
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  • #4
HallsofIvy said:
A sequence of real numbers is a "Cauchy sequence" if and only if |an- am| goes to 0 as m and n go to 0 independently: given [itex]\epsilon> 0[/itex] there exist N such that if m and n are both > N, then [itex]|a_n- a_m|< 0[/itex].
I think you meant as m and n go to infinity.
 
  • #5
Halls, I think you mean |a_n - a_m| < epsilon
 
  • #6
Yes, of course. I'll go back and edit so I can pretend I never made those mistakes!
 

FAQ: Is a Cauchy Sequence Always Convergent?

What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers where the distance between any two terms in the sequence becomes smaller and smaller as the sequence progresses. This means that the terms in the sequence eventually get closer and closer together.

How is a Cauchy sequence different from a convergent sequence?

A Cauchy sequence is a sequence that has terms that are getting closer and closer together, while a convergent sequence is a sequence that has a limit or a value that it approaches as the sequence progresses. While a Cauchy sequence is not guaranteed to have a limit, a convergent sequence always has a limit.

What is the importance of Cauchy sequences in mathematics?

Cauchy sequences are important in mathematics because they help us to define what it means for a sequence to be convergent. They also have many applications in real analysis and are used to prove important theorems, such as the completeness of the real numbers.

How can you determine if a sequence is Cauchy?

To determine if a sequence is Cauchy, you can use the Cauchy criterion, which states that a sequence is Cauchy if and only if for any positive real number ε, there exists a positive integer N such that for all m,n ≥ N, |an - am| < ε. In simpler terms, this means that the difference between any two terms in the sequence becomes arbitrarily small as the sequence progresses.

Can a Cauchy sequence be divergent?

Yes, it is possible for a Cauchy sequence to be divergent. This means that the sequence does not have a limit or a value that it approaches as the sequence progresses. However, in certain mathematical contexts, such as in complete metric spaces, all Cauchy sequences are guaranteed to be convergent.

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