Talk:Fortissimo Space is not Sequentially Compact

I've been looking at this and puzzling over this till I finally realised - of course, a sequence of integers only converges if it goes: $\ldots, k, k, k, \ldots$ for some $k \in \Z$. I raised the page Convergent Sequence in Set of Integers documenting this trivial piece of information, in the context of integers in the topological space induced by the metric space that is the real number line under the Euclidean metric. But it occurs to me that it has wider scope than this: a sequence of isolated points, perhaps, in ... what sort of space? I confess that I've posted up a lot of Steen and Seebach while not understanding all of it, so I might already have posted that proof up there.

But it's worth seeing whether such a result can be used in this page, which would save having to prove it again. --prime mover 16:23, 5 December 2011 (CST)

It is true for any discrete space. Even more, it is true not only for sequences, but also for nets (a generalitation of sequences).--Dan232 16:34, 5 December 2011 (CST)

Dan232 - thanks for your work on this. The tricky thing, as you've seen, is not so much working out the arguments, as being able to link them back to previously proved theorems. Finally I think we got there. --prime mover 14:28, 6 December 2011 (CST)