Talk:Combination Theorem for Sequences/Real

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)