Why are Cauchy sequences important in understanding limits and completeness?

[matqkks]

Why are Cauchy sequences important?
Is there only purpose to test convergence of sequences or do they have other applications?
Is there anything tangible about Cauchy sequences

 

[mathman]

http://en.wikipedia.org/wiki/Cauchy_sequence

Start here.

 

[mathwonk]

yes the purpose is to give a way to test whether a sequence is convergent without finding the limit.

 

[SteveL27]

yes the purpose is to give a way to test whether a sequence is convergent without finding the limit.

I always thought of them as a clever idea to define convergence even if we didn't have anything to converge to. For example in (0,1) the sequence {1/n} fails to converge ... but it "should" be a convergent sequence. It's the space that's deficient, not the sequence itself.

The concept of a Cauchy sequence formalizes that intuition. Then we can say that a space is complete if all the Cauchy sequences converge -- that is, if all the sequences that "should" converge, do converge.

So to me they're an important conceptual step in the process of understanding limits and completeness.

 

[chiro]

I always thought of them as a clever idea to define convergence even if we didn't have anything to converge to. For example in (0,1) the sequence {1/n} fails to converge ... but it "should" be a convergent sequence. It's the space that's deficient, not the sequence itself.

The concept of a Cauchy sequence formalizes that intuition. Then we can say that a space is complete if all the Cauchy sequences converge -- that is, if all the sequences that "should" converge, do converge.

So to me they're an important conceptual step in the process of understanding limits and completeness.

The thing though is that when you move to infinite-dimensional spaces, things get a bit weird.

The Cauchy sequences help with establishing some way of defining and managing ways of making sure that convergence exists in these environments.

If there wasn't these kinds of tools like the Cauchy sequences and related results then all the infinite-dimensional stuff would be like a house-of-cards that would most likely collapse and it can be hard for someone to appreciate if they haven't been exposed to the nature of infinite-dimensional geometry or vector space equivalents.