Talk:Limit of Subsequence equals Limit of Sequence

As Hausdorffness can occur without metrizability, the entire proof needs to be done with open sets in the topology. I will see what I can do. --Lord Farin 08:18, 10 October 2011 (CDT)


 * Feel free, and welcome. --prime mover 13:19, 10 October 2011 (CDT)


 * There seems to be no 'limit of sequence in Hausdorff space is unique' theorem. This appears a valuable addition. --Lord Farin 18:00, 11 October 2011 (CDT)


 * There definitely is - let me go find ... --prime mover 00:15, 12 October 2011 (CDT)
 * ... how does Convergent Sequence in Hausdorff Space has Unique Limit work for you - or is there a sublety I've missed? --prime mover 00:18, 12 October 2011 (CDT)


 * Proof of that is fine; thanks for pointing to it. However, your proof for $\R$ works with the standard (Borel) topology arising from the metric. There are a lot of other topologies on $\R$. That's why I removed it in the first place --Lord Farin 01:01, 12 October 2011 (CDT)


 * Okay, so put a mention about that fact. Please don't just delete stuff, please try and enhance it for accuracy. --prime mover 02:00, 12 October 2011 (CDT)

... Actually, thinking about it, we need two separate theorems: one for real numbers (purely in the lingo of Analysis) and one for the general Hausdorff space. The theorem for real numbers should then refer to the Hausdorff Space theorem, linking to the proof that the real number line under the standard (Euclidean) metric is Hausdorff. All the results are there - they just need to be knitted together. I'll get to it when I'm capable of rational thought again (heavy day at work today). --prime mover 13:44, 12 October 2011 (CDT)
 * Then in my opinion a theorem for Metric spaces should be formulated. The only difference is that $|x_n-l|<\epsilon$ should be replaced by $d(x_n,l)<\epsilon$. So this will yield practically free generalization. --Lord Farin 13:59, 12 October 2011 (CDT)
 * I had a feeling that might have been the original plan all along ... probably best linked together with a transclusion page. --prime mover 14:31, 12 October 2011 (CDT)

Proof of Hausdorff part
The second part of the proof doesn't seem quite right to me. I'll look into it during the coming weekend. JSchoone 15:30, 21 November 2011 (CST)


 * First of all: a last line saying, since U was arbitrary, we now have found that x_n_r converges to l.

Second: This result is true for even non-Hausdorff spaces. What goes wrong is that for this proposition it is not needed that limits are unique! Therefore, you can either add a note about uniqueness and Hausdorff, or remove the Hausdorff part.

If both of these are changed, or someone explains to me where I'm wrong, the proofread tag can be removed. JSchoone 08:46, 22 November 2011 (CST)
 * I agree; although the explaining note for the theorem speaks of 'the' limit, implying uniqueness. But nonetheless, the same argument applies to non-Hausdorff spaces. --Lord_Farin 09:09, 22 November 2011 (CST)

History of this page
The original proof was on the real line, and was designed for the analysis thread. However, Mr Big and Clever decided to make it as general as he could, and he read something about how it always converges in a Hausdorff space. That's before he knew much. (He still doesn't.) --prime mover 13:21, 22 November 2011 (CST)


 * I don't see where I'm wrong... JSchoone 14:29, 22 November 2011 (CST)


 * I'm not saying you are, it's just that the context of this page has evolved. Its original purpose was to point out that every subsequence of a convergent sequence in the reals converges on the same limit. With topologies which don't have unique limits, it's different. Just saying. If you don't like it, rename / delete / overwrite with something that you do like. --prime mover 14:37, 22 November 2011 (CST)