Talk:Combination Theorem for Sequences/Real

Analogous Statement for Euclidean Spaces
Is there an analogue of this already for Euclidean space? If not, then it follows directly from Limit of Sequence by Components (Euclidean Space) Andrew Salmon 21:25, 24 July 2012 (UTC)

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)