Talk:Convergent Sequence is Cauchy Sequence
A space in which every Cauchy sequence is convergent is called complete. Hence we should simply remark here that a space where the converse holds is called complete, with a reference to the definnition. And there should be a page "Real Number Line is Complete Metric Space", containing the second part of the proof.
By the way, I have a feeling there's already a result somewhere that a Cauchy sequence is convergent if and only if it has a convergent subsequence. Hence the fact that $\R$ is complete follows almost immediately from the Bolzano-Weierstraß Theorem.
We're moving house on Monday, so just taking a moment off from packing. If I need a break, I'll make these changes. -- lasserempe 12:13, 7 March 2009 (UTC)
Yeah yeah I know I know I'm getting round to it. I'm trying to negotiate the tricky tightrope between ensuring a rigorous link from the axioms of ZFC through to everything else, and I'm just being really careful not to mess up any of the linkages. That's why I get upset when stuff about specifics (e.g. results on real numbers) gets deleted in favour of a more general result (e.g. the same result on metric spaces and/or topologies) - because it may be that the specific result being generalised is one of those stepping-stones towards that general result. --prime mover (talk) 21:42, 7 March 2009 (UTC)
Oh yeah and I (basically) reverted this page back to where it was this morning, because I'm still working on it. Yes I know you can probably get rid of that page by proving it in the context of completeness but I want to follow this through myself in case I miss something. --prime mover (talk) 22:06, 7 March 2009 (UTC)