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soopo
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Homework Statement
What does it mean when a sequence is Cauchy?
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].
I think you meant as m and n go to infinity.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].
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.
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.
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.
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.
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.