Talk:Cauchy sequence
From Academic Kids
Cauchy net redirects here, yet there seems to be nothing about the concept here.... Vivacissamamente
- Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to converge. Nonetheless, Cauchy sequences do not always converge.
Some example please --Taw
added an example, it's kind of kludgy though -- RAE
I'd like to put something along the following lines: Cauchy Seqs are initially useful in spaces such as the Reals because they are a test of convergence which doesn't require a value for the potential limit. -- the flip side is that IF all CSs converge then a space is complete.
There's a sort of switch in perception as things move up a level of abstraction which as a mathematician I find self-evident (and interesting), but I suspect non-mathematicians find baffling or even terrifying:
- theorem: cauchy seqs converge on the Reals
- abstraction: cauchy seqs on other space, where they might not converge
- axiom: part of the defn of complete space
Has this been general idea been coverered anywhere in the maths section? -- Tarquin
I don't think it has been covered; it would fit either here or in complete space. AxelBoldt, Wednesday, June 12, 2002
how about:
- All Cauchy sequences of real or complex numbers converge, hence testing that a sequence is Cauchy is a test of convergence. This is more useful than using the definition of convergence, since that requires the possible limit to be known. With this idea in mind, a metric space in which all Cauchy sequences converge is called complete.
- Thus R and C are complete; but Q is not. The standard construction of the real numbers involves Cauchy sequences of rational numbers; (something about R being the completion of Q...)
...and something on Mathematical abstraction in general somewhere else. I'll see if I can dig up or remember the proof outlines for "Every convergent sequence is a Cauchy sequence" and "every Cauchy sequence is bounded" -- Tarquin June 12 2002
