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Mar8-12, 01:09 AM
P: 800
Cauchy Sequences

Quote Quote by mathwonk View Post
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.