Talk:Combination Theorem for Sequences/Real

Generalisation to series
Is it worthwhile to instantiate these theorems for the specific case of (absolutely convergent) series? --Lord_Farin 08:17, 27 April 2012 (EDT)
 * Why specifically absolutely convergent? --GFauxPas 08:21, 27 April 2012 (EDT)
 * A quick safeguard preventing me from saying things I am not sure about; but it might be that the imposition is unnecessary. Yes, coming to think of it I suspect it will hold regardless. But one has to tread carefully in this domain. --Lord_Farin 08:29, 27 April 2012 (EDT)
 * On the grounds that a series is itself a sequence (of partial sums) I would have thought that would already have been covered.
 * However, if it is indeed a definite fact that non-abs.conv.series don't behave, then my answer would be: Yes, go for it. --prime mover 08:33, 27 April 2012 (EDT)

The most important reason for bringing this up is that it is very common to use these theorems in the setting of a series, while one may (rightfully or not) wonder whether they are always directly applicable. Also, concerning my long-term plan to extend all of this stuff to incorporate 'diverging to infinity' in a formal $\overline\R$ sense, it will be convenient (especially in measure theory) to be able to directly refer to a statement on series.

On the long run, I suspect it might save doing the same boring exercise (or glossing over details) on a lot of pages. All in all, I suspect not that the proofs will be hard (I think they can use all of what is written here), but for the sake of rigour in dealing with these matters, I feel that it would be good. With that said, I am rather busy, so it will probably be a long-term project (of which I have already too many). --Lord_Farin 08:40, 27 April 2012 (EDT)

Generalisation to valued fields
I need these theorems for the general case of a valued field. This is to provide a link from the P-adic Norm not Complete on Rational Numbers. It doesn't seem to be necessary to instantiate these theorems for this case, but instead to change this page to the more general case. If this is the right approach then it may be better to change the page to cover normed division rings. Although the definition of a normed vector space seems to cover a normed division ring it seems a stretch to change this page to this, and in fact it seems preferable to create a separate page for normed vector spaces. What do others think? --Leigh.Samphier (talk) 06:56, 29 August 2018 (EDT)


 * I appreciate the thought that these often-used results could be generalised to such a generic application. One wonders, however, what is the best way to structure the pages so as to provide maximum reading comfort for all parties involved (being, at the extremes, both the expert needing full generality and the first-year undergrad student taking his first steps in $\R$)?
 * For, if your proposal is to be implemented without regard for those with less knowledge, I feel that we might risk the overall usefulness of the page. I would be happy to see some drafts before we decide anything. Would it be possible for you to create a proposal in your user space? &mdash; Lord_Farin (talk) 13:37, 29 August 2018 (EDT)